Обобщённые инварианты Хованова узлов в прямоугольных представлениях тема диссертации и автореферата по ВАК РФ 01.01.06, кандидат наук Кононов Яков Александрович

• Кононов Яков Александрович
• кандидат наук
• 2020

Оглавление диссертации кандидат наук Кононов Яков Александрович

1.1 Суммы по путям

1.2 Формула Россо-Джонса

1.3 Суперполиномы БАНА

2 Свойства факторизации

2.1 Предел в q =1

2.2 Предел в корнях из единицы

2.3 Многочлен Александера

2.4 Многочлены Кашаева

3 Дифференциальное разложение и дефект

4 Обобщённые инварианты Хованова твистованных узлов

4.1 Локализация

4.2 Узел-восьмёрка

4.3 Другие твистованные узлы

4.4 Сравнение нашего подхода к обобщённым инвариантам Хованова с другими

Список литературы

Приложение А. Статья 1. Yakov Kononov, Alexei Morozov "On the defect and stability of differential expansion"

Приложение Б. Статья 2. Yakov Kononov, Alexei Morozov "Factorization of colored knot polynomials at roots of unity"

Приложение В. Статья 3. Yakov Kononov, Alexei Morozov "Colored HOMFLY and generalized Mandelbrot set"

Приложение Г. Статья 4. Yakov Kononov, Alexei Morozov "On factorization of generalized Macdonald polynomials"

Приложение Д. Статья 5. Yakov Kononov, Alexei Morozov "Rectangular superpolynomials for the figure-eight knot 4i"

Приложение Е. Статья 6. Yakov Kononov, Alexei Morozov "On rectangular HOMFLY for twist knots".

Введение диссертации (часть автореферата) на тему «Обобщённые инварианты Хованова узлов в прямоугольных представлениях»

Введение

Первый квантовый инвариант узлов и зацеплений был введён Джонсом [21] в 1984. Инвариант был построен в терминах skein relations, имеющих теоретико-представленческий смысл.

(q - q

-,—2

2

q

Позднее Решетихин и Тураев [22] обобщили конструкцию на случай узлов в 3-мерных многообразиях с произвольными группами и представлениями вместо фундаментального представления sl(2). Важнейшей задачей было нахождение описания инварианта Решетихина-Тураева, не использующего двумерную проекцию. Виттен [26] в 1989 показал, что инвариант Решетихина-Тураева связан с наблюдаемыми в теории Черна-Саймонса. Теория Черна-Саймонса (CS) - это калибровочная теория с действием

S = j AdA +2A3,

где A - связность в главном G-расслоении над 3-мерным многообразием для компактной группы Ли G. Виттен связал skein relation с тем, что пространство некоторых конформных блоков двумерно.

Полином HOMFLY H\(A, q) определяется физически как коррелятор голономии связности в представлении А группы G с мерой e-S. Наиболее интересный пример отвечает случаю G = SU(N). В этом случае представления параметризуются диаграммами Юнга А = [Ai, А2, ...А^д)], и мы будем обозначать представление и диаграмму Юнга одинаково. Оказывается, что полиномы HOMFLY являются, действительно, многочленами Лорана от переменных A и q, где

q2 = exp (-— ) , A = qN, g = the coupling constant.

\g + NJ

Поскольку квантовые инварианты узлов оказываются многочленами Лорана с целыми коэффициентами, естественная задача - интерпретировать их как градуированные характеры некоторых векторных про-

странств (гомологий), ассоциированных с узлами и зацеплениями. Хо-ванов [9] в 1999 открыл так называемые гомологии Хованова, которые дают категорификацию полинома Джонса.

1 Вычисление цветных многочленов ИОМЕЬУ 1.1 Суммы по путям

По теореме Артина, каждый узел может быть представлен как замыкание косы. Простейший и прямолинейный способ вычисления цветных инвариантов ИОМЕЬУ узлов состоит в вычислении свёрток И,-матриц [6].

Рис. 1: Узел, представленный как замыкание комы. Диаграмма отвечает произведению К\2К23.

Каждой нити в косе сопоставляется представление Л, а нескольким параллельно идущим нитям сопоставляется тензорное произведение представлений. Многочлен ИОМЕЬУ равен следу произведения И,-матриц Яг,г+1 для каждого перекрёстка в косе. Проще всего описать И,-матрицы в случае векторного представления Л = □ группы БЬ(Ы). Более общий случай произвольного представления может быть сведён к случаю векторного представления. Тензорное произведение векторных представлений имеет следующее разложение на неприводимые представления:

□ ®п = м [Лр^Т(А)| |А|=п

где кратности - числа стандартных таблиц Юнга с данной формой. Обозначим через Ут базисный элемент (Гельфанда-Цейтлина), отвечающий таблице Т. Действие операторов Яг,г+1 в этом базисе устроено следующим

образом. Рассмотрим таблицу Т0, получанную применением элементарной транспозиции (г, г + 1) к Т. Возможны два случая: когда полученная таблица является стандартной таблицей Юнга, и когда нет.

1. Если Т0 не является стандартной таблицей Юнга, то Ут - собственный вектор для Я^+ь

2. Если Т0 является стандартной таблицей Юнга, то мы имеем блок 2

где п - разность значени содержаний клеток г и г + 1.

Эти формулы являются ^-деформацией формул для действия симметрической группы в базисе Юнга, полученных А.Окуньковым и А.Вершиком в [1]. В частности, оператор действует всегда диагонально в этом базисе, так как "1"не может быть переставлены ни с какой другой клеткой диаграммы.

Таким образом, многочлен ИОМПУ в векторном представлении является суммой по путям в графе Юнга. В графе Юнга вершины параметризуются неприводимыми представлениями. Две вершины соединены ребром, если одна является подпредставлением другой, тензорно умноженной на векторное представление.

Для произвольного представления Л с п клетками, необходимо рассмотреть "каблирование"(замена каждой нити на п нитей), в соответствии со стандартной теорией.

;т, г, г + 1 в одной строке Л д-1ут, г, г + 1 в одном столбце Л

на 2:

1

2

3

Рис. 2: Doubling

1.2 Формула Россо-Джонса

Один из простейших типов узлов - торические узлы Т(п,т). Они параметризуются парами положительных взаимно-простых чисел. Инварианты ИОМЕЬУ таких узлов были вычислены Россо и Джонсом в знаменитой статье [23], и формула для многочлена ИОМЕЬУ в представлении Л торического узла известна как формула Россо-Джонса:

V-{А}

На(Т (п,т))= (д т С Фт^ где Эа суть полиномы Шура,

Фт : Рк ! Ртк

- операция Адамса, и Ш2 - так называемый оператор разрезания и склейки, который действует диагонально в базисе многочленов Шура:

^2ЭА = (X №) - *'(□))) ЭА-

\КА /

Здесь а',1' - длины «корук» и «коног», соответственно. Мы также пользуемся обозначениями

{х} = х - х-1, [п] = Щ-, [п]! = Ц[г].

г—1

{д}

При специализации

= {Аг} А = N Рг = {дг} , А = д

многочлены Шура Эа дают квантовые размерности неприводимых представлений группы Цд(д[(У)):

-г-г N + а - V]

ЭА!П ТГТОГТГ

г

a

a

l

Рис. 3: Определение о!,1',а,1 1.3 Суперполиномы ВЛИЛ

Существует однопараметрическая деформация формулы Россо-Джонса [3], которая называется суперполиномом ВАНА. Вместо Фоковского пространства симметрических функций, необходимо рассмотреть К-теорию схемы Гильберта точек на плоскости С2 , которая отождествляется с пространством Фока посредством конструкций [19, 20].

V

(Д^Ли }|PraAm|Ovir)

K(Hilb(C2,m|A|))

где - полином Макдональда от генераторов алгебры Гейзенберга

наклона n/m в квантовой группе U~(qI(1)), действующей на К-теории схемы Гильберта, а U и Ovir - тавтологический и структурный пучки.

Гипотетически [4] каждой косе на n нитях сопоставляется пучок на схеме Гильберта n точек, эйлерова характеристика которого даёт суперполином некрашенного узла или зацепления.

2 Свойства факторизации

2.1 Предел в q = 1

Центральное тождество в теории цветных многочленов HOMFLY - факторизация в точке q =1:

Hx(A,q =1)= (Hn(A,q =1}) |Л' (1)

У этого тождества бесконечно много различных обобщений. С точки зрения матричных моделей, оно отвечает специальному пределу

q = e~, ~ ! 0, N~ = const,

и имеет глубокие связи с теорией Гурвица. Заметим, что суперполиномы DAHA гипотетически обладают более слабым факторизационным свойством

P[r](q = 1,t, A) = (pn(q = 1,t,A})r P[ir](q = 1,t, A) = (pn(q = 1,t,A})r

Здесь [r] и [1r] отвечают симметрическому и антисимметрическому представлению, соответственно.

2.2 Предел в корнях из единицы

В работе [13] получены гипотезы о факторизации в других корнях из единицы. Когда q2m = 1, для симметрических представлений верно:

H[n+m] = H[n] ' H[m] •

Для более общих представлений, гипотетически

H = HЛ ■ HS

'm ,

если ^ получен приклеиванием косого крюка длины т к Л. Другими словами, ^/Л - связная косая диаграмма Юнга ширины 1. Это в точности диаграммы, возникающие в правой части правила Мурнагана-Накаямы

'-1 V

Ожидается, что при правильном обобщении этих гипотез возможен прорыв в понимании А-полиномов - рекуррентных разностных соотношений на цветные многочлены узлов.

Pm ■ SЛ = X(-1)lengthW^-1 s

Рис. 4: Пример факторизации #[5,4,4,3,3,2,1,1] = #[5,3,2,2,1] ■ #[10], где q =

ехР (то)

2.3 Многочлен Александера

Многочлен Александера определяется как

А1д(^ = #л(А =1,q). Для представлений с единственным крюком

/ ЛИ

A1л(q) = (аШ] . (2)

2.4 Многочлены Кашаева

Полиномы Кашаева определяются как специализация цветных многочленов НОМПУ в примитивных корнях из единицы:

Кл(А) = #л ^q = ехр -щ) , Л^ .

Согласно нашей гипотезе [13], когда Л - диаграмма с единственным крюком,

Кл(Л) = К^ (А1л1) .

Вместе с (2) это обозначает некоторую A-q дуальность, которая является нелинейным обобщением дуальности между уровнем и рангом в квантовых аффинных группах.

3 Дифференциальное разложение и дефект

Для симметрических представлений [г] (отвечающих диаграммам Юнга с одной строкой из г клеток) многочлены НОМПУ обладают следующим разложением

Г |- -| 8—1

#[г] = 1 + X Г С-(А, q){A/q} П^г+7'}.

8=1

.7=0

Здесь С8(Л, q) - полиномы Лорана, не зависящие от г, а q-биномиальный коэффициент определяется как

[г]!

[в]![г - в]!'

Коэффициенты 08 можно вычислять рекуррентно, зная многочлены НОМПУ в симметрических представлениях.

Эта структура является q-деформацией свойства (1):

(#[1])|Л|

#л|

Л \п=1

9=1

Данное разложение следует из следующих трёх свойств многочленов НОМПУ [11]:

1.

2. Когда А = qr+s,

3. Когда А = q

1

#л* (А, q) = #л(А, 1/д),

#[1Г ] = #[р].

#[1Г ] = 1.

Согласно нашей гипотезе [12], если для какого-то узла 08 делится на некоторые факторы {Aqfc}, то это же верно для 0'3 при в' > в. Более того, делятся на некоторые факторы {Aqfc} в соответствии с очень чётким законом, который описывается для каждого узла единственным

параметром 8 > 0. Именно, делится на

[ ^ ]

Оказывается, что дефект 8 - это очень простая величина - она равна степени полинома Александера

8 = ^„2 Н(А, д)

А=1

Интересно, что существуют узлы с тривиальным полиномом Александера, и для них все делятся на единственный множитель { А}.

Ад9 Ад8 Ад7 Ад6 Ад5 Ад4 Ад3 Ад2 Ад1 Ад0

Ад4 Ад3 Ад2 Ад1 Ад0

23456789 10 11

3 4 5 6 7 8 9 10 11 12

Рис. 5: Делимость для 8 =1 слева, для 8 = 2 справа.

Очень интересным следствием этих свойств является то, что для любого заданного р > 0, при специализации А = д-р, последовательность коэффициентов

Н(А, д)

по д стабилизируется. Для иллюстрации, рассмотрим узел 62, полином Александера которого равен

А/(б2) = -3 - д-4 + 3 д-2 + 3д2 - д

4

Н1 1

Н2 1 - 3 92 + д4 + 5 д6 - 8 98 + 3 д10 + 10 д12 - 10 д14 - 4 д16 + 9 д18 - д20 - 3 д22 + д24

Нз 1 - 3 д4 - 3 д6 + 4 д8 + 9 д10 - 2 д12 - 12 д14 - 6 д16 + 11 д18 + 14 д20 - 2 д22 - 12 д24- -8 д26 + 4 д28 + 9 д30 + 2 д32 - 3 д34 - 3 д36 + д40

Н4 1 - 3 д6 - 3 д8 + 4 д12 + 9 д14 + 2 д16 - 4 д18 - 12 д20 - 8 д22 + 2 д24 + 11 д26 + 14 д28+ +2 д30 - 4 932 - 12 934 - 8 д36 + 4 д40 + 9 д42 + 2 д44 - 3 д48 - 3 д50 + д56

Н5 1 - 3 д8 - 3 д10 + 4 д16 + 9 д18 + 2 д20 - 4 д24 - 12 д26 - 8 д28 + 2 д32 + 11 д34 + 14 д36+ +2 д38 - 4 д42 - 12 д44 - 8 д46 + 4 д52 + 9 д54 + 2 д56 - 3 д62 - 3 д64 + д72

Нб 1 - 3 д10 - 3 д12 + 4 д20 + 9 д22 + 2 д24 - 4 д30 - 12 д32 - 8 д34 + 2 д40 + 11 д42 + 14 д44+ +2 д46 - 4 д52 - 12 д54 - 8 д56 + 4 д64 + 9 д66 + 2 д68 - 3 д76 - 3 д78 + д88

Таким образом, при фиксированной специализации А = q—k коэффициенты инвариантов симметрических представлений стабилизируются и не несут дополнительной топологической информации.

4 Обобщённые инварианты Хованова твисто-ванных узлов

4.1 Локализация

Пусть X - гладкое алгебраическое многообразие с действием алгебраического тора Т, для которого число неподвижных точек конечно. Тогда эйлерова характеристика эквивариантного пучка & на X может быть вычислена как

где Xт - подмногообразие неподвижных точек, и т - его нормальное расслоение.

Наиболее интересный пример для нас - схема Гильберта точек на плоскости С2. Это многообразие Накаджимы, отвечающий колчану (рис.

Явно, Ы1Ь(С2,п) определяется как множество идеалов I С С[х,у] конечной коразмерности п. Двумерный тор действует следующим образом:

Неподвижные точки отвечают диаграммам Юнга размера п. Есть универсальное расслоение и со слоем С[х, у]/1 над идеалом I. Характер

X(X, F) = x(XT,F ® т),

6).

Рис. 6: Jordan quiver

X

X

X

X

ху

2

х2 у

У2

ху2

У3

ху3

У4

1

У

Рис. 7: Неподвижные точки в ИПЪ(С2). Диаграмма отвечает идеалу, по-

5 3 2 2 4 5

рожденному х5, х3у, х2у2, ху4, у5.

касательного пространства относительно действия тора имеет следующее описание в терминах "рук"и "ног"клеток диаграммы:

□ел

4.2 Узел-восьмёрка

Рис. 8: Узел-восёмёрка слева. Твистованный узел с шестью перекрутками справа

С точки зрения дифференциального разложения, одними из наиболее простых узлов являются твистованные узлы, а среди них наиболее простой - узел-восьмёрка. Для узла-восьмёрки 41 в прямоугольных пред-

ставлениях следующая формула была получена нами в [16]:

Нл = X П КЗуС'"'' {Лх/-»'}{Ау-у'-»'},

л □ел {Я }

где

х = яг, у = Я*.

Из-за присутствия множителей {хя"'~1'}{уя''-а'} в числителе, при специализации х = я", у = Я^ сумма ограничивается диаграммами, которые помещаются в прямоугольник г х 5.

Данная формула имеет простой смысл в исчислительной геометрии: это локализационное вычисление эйлеровой характеристики следующего пучка на ИИЪ(С2) для действия с весами ¿1 = я2, ¿2 = Я-2:

Д0 (х (и + Аи_)) 0 Д° (у-1 (и + А-1и_)) 0 К1г/2.

Здесь Д0 обозначает симметризованную версию внешней алгебры

Д> ) = Д>) 0 )-1/2,

и Кчг = ¿е^Т^г)-1 - виртуальный канонический пучок.

Это мотивирует деформацию конструкции - рассмотреть эквивари-антную эйлерову характеристику этого же пучка на Ы1Ъ(С2) с произвольными весами ¿1 = я2,^2 = ¿-2.

Для узла-трилистника, необходима незначительная модификация пучка

(-А2яД)х0ае1(и )-10Д0 (х (и + Аи _))0Д0 (у-1 (и + А-1и _ ))0К-1Л 4.3 Другие твистованные узлы

Твистованные узлы параметризуются целым числом т 2 Ъ, так что т = 1 отвечает узлу-восьмерки, т = —1 - трилистнику, и т = 0 -незаузленности. Мы также получили подобные [17] (но более сложные) формулы для других твистованных узлов. Наши формулы были переформулированы и открыли новую страницу для исследований в [27, 28].

4.4 Сравнение нашего подхода к обобщённым инвариантам Хованова с другими

Данные результаты открыли новую страницу исследований. Это первый пример описания суперполиномов не итерированно-торического узла в большом количестве нетривиальных представлений. Единственным пересечением семейств твистованных и торических узлов является трилистник. Можно сравнить две конструкции гипотетических инвариантов Хованова этого узла: наше выражение как для твистованного узла в терминах дифференциального разложения, и как для торического узла в терминах действия квантовой тороидальной алгебры Ц~(£|[(1)). Во всех вычисленных примерах оба подхода дают одинаковые ответы, что удивительно. В обеих теориях возникает геометрия схемы Гильберта, но в совершенно разных аспектах. Также гипотетически наши многочлены совпадают с многочленами Пуанкаре гомологий ИОМПУ, построенными Ховановым и Рожанским для фундаментального представления [5].

Результаты диссертации опубликованы в шести статьях:

1. Yakov Kononov, Alexei Morozov «On the defect and stability of differential expansion», Journal of Experimental and Theoretical Physics Letters, 2015, 101:12, 8

2. Yakov Kononov, Alexei Morozov «Factorization of colored knot polynomials at roots of unity», Physics Letters B, Volume 747, 30 July 2015, Pages 500-510

3. Yakov Kononov, Alexei Morozov «Colored HOMFLY and generalized Mandelbrot set», Journal of High Energy Physics, volume 2015, Article number: 151 (2015)

4. Yakov Kononov, Alexei Morozov «On factorization of generalized Macdonald polynomials», Eur. Phys. J. C (2016) 76:424

5. Yakov Kononov, Alexei Morozov «Rectangular superpolynomials for the figure-eight knot 4i», Theoretical and Mathematical Physics, volume 193, pages 1630-1646(2017)

6. Yakov Kononov, Alexei Morozov «On rectangular HOMFLY for twist knots», Mod.Phys.Lett. A Vol. 31, No. 38 (2016) 1650223

Список литературы

[1] A. Okounkov, A. Vershik, New approach to representation theory of symmetric groups, POMI letters, 2004, volume 307, 57-98.

[2] A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363.

[3] E. Gorsky, A. Negut, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9) 104 (2015), no. 3, 403-435

[4] E. Gorsky, A. Negut, J. Rasmussen, Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology, arXiv:1608.07308

[5] M. Khovanov, L. Rozansky, Matrix factorizations and link homology, arXiv:math/0401268

[6] A. Anokhina, A. Mironov, A. Morozov, An. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Advances in High Energy Physics, Volume 2013 (2013) 931830

[7] A. Okounkov, R. Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675-699

[8] N.M. Dunfield, S. Gukov, J. Rasmussen, The Superpolynomial for Knot Homologies, Duke Math. J. 101 (2000), no. 3, 359-426

[9] M. Khovanov, A categorification of the Jones polynomial,Duke Math. J. 101 (2000), no. 3, 359-426

[10] M. Khovanov, Universal construction of topological theories in two dimensions, arXiv:2007.03361

[11] Ya. Kononov, Colored HOMFLY polynomials, Thesis at the Independent University of Moscow

[12] Ya. Kononov, A. Morozov, On the defect and stability of differential expansion, JETP Letters 101 (2015) 831-834

[13] Ya. Kononov, A. Morozov, Factorization of colored knot polynomials at roots of unity, Phys.Lett. B747 (2015) 500-510

[14] Ya. Kononov, A. Morozov, Colored HOMFLY and Generalized Mandelbrot set, JHEP 1511 (2015) 151

[15] Ya. Kononov, A. Morozov, On Factorization of Generalized Macdonald Polynomials, Eur.Phys.J. C76 (2016) no.8, 424

[16] Ya. Kononov, A. Morozov, Rectangular superpolynomials for the figure-eight knot, Theoretical and Mathematical Physics 193 (2017) 1630-1646

[17] Ya. Kononov, A. Morozov, On rectangular HOMFLY for twist knots, Mod.Phys.Lett. A Vol. 31, No. 38 (2016) 1650223

[18] I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195

[19] T. Bridgeland, A. King and M. Reid , The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14

(2001), 535-554

[20] M. Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current Developments in Mathematics, Volume 2002

(2002), 39-111.

[21] V. Jones, A polynomial invariant for knots via von Neumann algebra, Bulletin of the American Mathematical Society. (N.S.). 12: 103-111

[22] N. Reshetikhin and V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys., Volume 127, Number 1 (1990), 1-26.

[23] M. Rosso and V. Jones, , On the invariants of torus knots derived from quantum groups, J. Knot Theory Ramifications, 2 (1993) 97-112

[24] S. Chern, J. Simons, Characteristic forms and geometric invariants, Annals of Mathematics. 99 (1): 48-69

[25] L.-H. Robert and E. Wagner, A closed formula for the evaluation of

-foams, to appear in Quantum Topology, arXiv:1702.04140.

[26] E. Witten, Quantum Field Theory and the Jones Polynomial, Communications in Mathematical Physics. 121 (3): 351-399

[27] M. Kameyama, S. Nawata, R. Tao, H.D. Zhang, Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams, arXiv:1902.02275

[28] L. Bishler, A. Morozov, Perspectives of differential expansion, arXiv:2006.01190

Приложение А

Статья 1. Yakov Kononov, Alexei Morozov "On

the defect and stability of differential expansion".

Journal of Experimental and Theoretical Physics Letters, 2015, 101:12, 8

Разрешение на копирование: Согласно Соглашению о копирайте автор статьи может использовать полную журнальную версию статьи в своей диссертации при условии, что указан источник.

ISSN 0021-3640, JETP Letters, 2015, Vol. 101, No. 12, pp. 831-834. © Pleiades Publishing, Inc., 2015.

On the Defect and Stability of Differential Expansion^

Ya. Kononovd and A. Morozova-c

a Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia e-mail: [e-mail] b National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia c Institute for Information Transmission Problems, Moscow, 127994 Russia d Math Department, Higher School of Economics, Moscow, 117312 Russia Received April 30, 2015

Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern—Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in rth symmetric representation does not change with r, if it is large enough. This fact reflects the non-trivial and previously unknown properties of the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.

DOI: 10.1134/S0021364015120127

1. INTRODUCTION

HOMFLY polynomials are Wilson-loop averages in 3 d Chern—Simons theory [1], which in this simplest model depend only on the topology of the Wilson line (knot). Therefore, one can separate and study the group-theory properties of observables; this is a nontrivial and very interesting problem, for a brief summary of results see [2]. From the quantum field theory perspective knot polynomials are direct generalization of conformal blocks, and this relation [3] provides one of the effective calculation methods in knot theory.

Recent advances in [4, 5], based on the previous considerations in [3—15], provided a way to systematically calculate simplest colored HOMFLY polynomials [16] for a really wide variety of knots including, in particular, the entire Rolfsen table of [17]. This allows us to return to the study of "differential expansions" of [18—27], which was temporarily postponed because of the insufficient "experimental" material.

In this work, we describe empirically obtained properties of these expansions for symmetric representations [r] (where r is the length of the single-line Young diagram). It looks like there are different uni-versality classes of such expansions, characterized by a single integer, which we call "defect" 83£. Moreover, these newly observed properties allow identifying 2(83i + 1) with the power of the Alexander polynomial and lead to a peculiar stability property of symmetrically colored HOMFLY for large enough r: what stabilizes is not the polynomial itself, but the set of its coefficients, i.e., something like the "coordinates" gr j, introduced in [24]. Theoretical analysis of these obser-

vations, proofs and extension to non-(anti)symmetric representations are beyond the scope of the present text.

2. NOTION OF DEFECT

Differential expansion provides a knot-dependent q-deformation (quantization) of the remarkable factorization property [9, 11, 19—21] of colored "special" polynomials at q =1,

JR||

hR(A) = (H1](A))'J = !

V representation R and knot

(!)

which fully defines their dependence on representation (Young diagram) R. Currently these expansions can be well studied only for symmetrically-colored HOMFLY, and we focus on this case in the present paper. The story starts from the fact that

• Hr = H[r] always possesses differential expansion of the following form:

B? (A, q2 ) = !

+

I

s = !

[r] !

G?(A, q){A/q}^{Aq + j}.

s - !

[ s ] ! [ r - s ] !

(2)

j = o

^The article is published in the original.

For generic knot Gs is a non- factorizable Laurent polynomial oL4 and q, but for some knots it can be further factorized. In this formula we use the notation {x} = x — x-1 and quantum number is defined as [n] =

{qn}/{q};

r

what is important, if Gs is divisible by some "dif-

ferential" {Aqk}, the same is true for all other Gy , with s' > s. This property allows one to introduce defect

. • fC fC -f fC

functions vs and = s — 1 — vs :

vf - 1

Gf (A, q) = Ff(A, q) ^ {Aj

j = o

2 f s - 2 - ^s

(3)

= Ff (A, q) n {A4 }

j = o

f . f f . f o

Vs ^ V,. , | ^ Is' for S < S ,

(4)

3. RELATION TO THE ALEXANDER POLYNOMIAL

It is an interesting question, if the value of Sf can also restrict the coefficient functions Ff (A, q).

Immediately observable are two remarkable properties of this kind:

• Gf (A, q) has power 2Sf in q2, i.e., Gf (A, q) =

SL 8* cjqX

f

which are both (!) monotonically increasing function of s,

For example, Sf = 0 whenever Gf is independent of q;

• Alexander polynomial has power 2(Sf + 1) in q2,

i.e., AP\q) = HV =1, q)= Y8* +1 f

¿Si = s-1

i.e., both grow, but not faster than s;

• for A = qNwith any fixed N, positive or negative,

C

qN, q{q}"s ^(qN, q) - {q}s-1, (5)

i.e., the sth term of differential expansion at fixed N is actually of the order {q}2s;

'j = -

f

2J

sf = - Power 2(At )- 1. 2 q

(12)

• it turns out that vs as a function of s has a very special shape, fully parameterized by a single integer Sf > —1, which we call the defect of differential expansion:

defect Sf = -1 ^ |f = s- 2, vf = 1, (6) defect Sf = 0 ^ |f = 0, vf = s- 1, (7)

defect Sf = 1 => |f ~ 2, vf = entier(s---1-J , (8)

f f 2 s f ( s_11

defect S = 2 ^ |s~ -3-, Vs = entier^ --3- J, (9)

f f 3 s f ( s_1J

defect S = 3 => |s ~ —, vs = entier( —4—J, (10)

For Sf ^ 0 these facts are not immediately related:

f

contributing to Alexander polynomials are all Gs with

s < Sf + 1 and they can and do contain much higher powers in q. Moreover, even the product of differentials in the s-term has power in q, which grows quadratically with s and thus with Sf. This means that there are serious cancelations behind the linear law (12).

Since Alexander polynomials are easily available already from [17], the values of v for each knot are easily obtained from this data.

4. TWIST AND TORUS KNOTS For all twist knots the defect is vanishing

8twist = 0. (13)

Instead of torus knots, it is a kind of maximal:

for the 2-strand family S[2' n] = n ^ 3 , for the 3-strand family

(8№ 10124> •••)

o[3, n] 0

S ] = n - 2 ,

for the 4-strand family S

In general

[4, n] = 3 n - 5 = 2 ,

f .. ( s- 1

Vs = entier( Z-^

Sf + 1y Sf + 1

f

f

| s = s 1 V

f

Sf + 1

-s.

(11)

in general

g[m, n] = mn - m - n - 1 (14)

since the power of the Alexander polynomial A/[m, n] is (m - 1)(n - 1).

JETP LETTERS Vol. 101 No. 12 2015

ON THE DEFECT AND STABILITY OF DIFFERENTIAL EXPANSION

833

5. NEGATIVE DEFECT: KTC MUTANTS AND THEIR RELATIVES

Starting from 11 intersections there are cases when the Alexander polynomial is just unity; i.e., the defect

is negative, 8% = -1. According to our general rules,

%

this means that for such knots already G1 is reducible:

% %

G1 ~ {4}. Of course, also all other Gs ~ {A}, because all the terms of the differential expansion are vanishing for A = 1.

This is indeed true for the first example—the celebrated Kinoshita-Terasaka and Conway (KTC) mutants % = 11n42 & 11n34, reconsidered recently in

[5]—and for the next example, available from [17]: % = 12n313 & 12n430. Moreover, the combination of [4] and [5] allows to calculate HOMFLY for KTC mutants for any symmetric representation and validate

(6) in this particular example.

6. SUMMARY: STABILITY AND OTHER PROPERTIES OF DIFFERENTIAL EXPANSION

Take any randomly chosen knot (say, % = 62).

It is easy to observe that, starting from H|2, the sets of coefficients are the same, despite the polynomials are different. At A = q-2 the same happens, beginning

from . Thus, what stabilizes are not the polynomials themselves, but something else, more appropriately associated with the knots. In full accordance with the

vision in [24] this something are the coefficient functions

%

Gs of the differential expansion.

Due to their properties, which are revealed in the present paper, contributing at A = q-N are just the first few terms of the expansion (2):

B? (A = !, q ) = H? (A = !, q)

(!6)

H?(A = !

N

q

(N +1)(s + DrÄr , N

= ! - £ [ N + ! ] G?(A = q )

s = o

s - !

n{q

j = o

{q }s - ! [ s ] !

(!5)

r - N+j

}{q -1},

where the last product is Laurent polynomial in qr and due to (5) the ratio in front of it is an r-independent polynomial. Thus what we get is just a sum of a few polynomials, multiplied by different powers of qr. They do not overlap at large enough r, and this provides an r-independent set of the coefficients, as in the above example.

In fact, one could wish to interpret the remarkable identity [19, 20]

for Alexander polynomials as a manifestation of the same phenomenon at N = 0. However, this is literally so only for 8% = -1 and 8% = 0. Still (16) is true not only for all knots, but actually for all single-hook (and not just single-line) representations R. For such representations (16) is a kind of a dual to (1).

7. CONCLUSIONS

We have studied the "quality" of the differential expansion (2) for symmetrically colored reduced HOMFLY polynomials, which are the typical observables in the simplest possible Yang—Mills theory. If only naive representation-theory properties are taken into account from (1) to restriction l < N on the number l of lines in the Young diagram for particular SL(N), this expansion has the form (2) with irreducible polynomial coefficient functions Gs (A, q). It is well known, however, that sometime Gs are further fac-torized, thus adding more restrictions/structures to the color-dependence of physical observables. Now, when methods were developed to study entire classes of generic knots, we could attack this problem in a systematic way and demonstrate that Gs are always factor-izable for high enough s. The depth of factorization appeared to depend on a single characteristic of the knot, which we originally called defect of the expansion, and further demonstrated that it is linearly related to the degree of Alexander polynomial, what makes it very easy to find.

This factorization universality leads to remarkable kind of stabilization of symmetrically colored HOMFLY, ensuring that increasing r beyond some knot-dependent boundary does not provide new physical (topological) information. This is what one naturally expects, and now we see how this actually works.

Highly desirable is extension of this new insight beyond pure symmetric and antisymmetric representations, but this requires further development of technical tools in conformal, quantum group and ^-matrix theories.

This work was supported in part by the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project no. NSh-1500.2014.2); by the Russian Foundation for Basic Research (project no. 13-02-00478); by the joint grant nos. 15-52-50041-YaF, 14-01-92691-Ind-a, 15-51-52031-NSC-a; by the Brazilian Ministry of Science, Technology and Innovation through the National Counsel of Scientific and Technological Development; and by the Laboratories of Algebraic Geometry and Mathematical Physics, Higher School of Economics.

JETP LETTERS Vol. !0! No. !2 20!5

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JETP LETTERS Vol. 101 No. 12 2015

Приложение Б

Статья 2. Yakov Kononov, Alexei Morozov "Factorization of colored knot polynomials at roots of unity".

Physics Letters B, Volume 747, 30 July 2015, Pages 500-510

Разрешение на копирование: Согласно Соглашению о копирайте автор статьи может использовать полную журнальную версию статьи в своей диссертации при условии, что указан источник.

Список литературы диссертационного исследования кандидат наук Кононов Яков Александрович, 2020 год

Список литературы

[1] A. Okounkov, A. Vershik, New approach to representation theory of symmetric groups, POMI letters, 2004, volume 307, 57-98.

[2] A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363.

[3] E. Gorsky, A. Negut, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl. (9) 104 (2015), no. 3, 403-435

[4] E. Gorsky, A. Negut, J. Rasmussen, Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology, arXiv:1608.07308

[5] M. Khovanov, L. Rozansky, Matrix factorizations and link homology, arXiv:math/0401268

[6] A. Anokhina, A. Mironov, A. Morozov, An. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Advances in High Energy Physics, Volume 2013 (2013) 931830

[7] A. Okounkov, R. Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675-699

[8] N.M. Dunfield, S. Gukov, J. Rasmussen, The Superpolynomial for Knot Homologies, Duke Math. J. 101 (2000), no. 3, 359-426

[9] M. Khovanov, A categorification of the Jones polynomial,Duke Math. J. 101 (2000), no. 3, 359-426

[10] M. Khovanov, Universal construction of topological theories in two dimensions, arXiv:2007.03361

[11] Ya. Kononov, Colored HOMFLY polynomials, Thesis at the Independent University of Moscow

[12] Ya. Kononov, A. Morozov, On the defect and stability of differential expansion, JETP Letters 101 (2015) 831-834

[13] Ya. Kononov, A. Morozov, Factorization of colored knot polynomials at roots of unity, Phys.Lett. B747 (2015) 500-510

[14] Ya. Kononov, A. Morozov, Colored HOMFLY and Generalized Mandelbrot set, JHEP 1511 (2015) 151

[15] Ya. Kononov, A. Morozov, On Factorization of Generalized Macdonald Polynomials, Eur.Phys.J. C76 (2016) no.8, 424

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[17] Ya. Kononov, A. Morozov, On rectangular HOMFLY for twist knots, Mod.Phys.Lett. A Vol. 31, No. 38 (2016) 1650223

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Приложение А

Статья 1. Yakov Kononov, Alexei Morozov "On

the defect and stability of differential expansion".

Journal of Experimental and Theoretical Physics Letters, 2015, 101:12, 8

Разрешение на копирование: Согласно Соглашению о копирайте автор статьи может использовать полную журнальную версию статьи в своей диссертации при условии, что указан источник.

ISSN 0021-3640, JETP Letters, 2015, Vol. 101, No. 12, pp. 831-834. © Pleiades Publishing, Inc., 2015.

On the Defect and Stability of Differential Expansion^

Ya. Kononovd and A. Morozova-c

a Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia e-mail: [e-mail] b National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia c Institute for Information Transmission Problems, Moscow, 127994 Russia d Math Department, Higher School of Economics, Moscow, 117312 Russia Received April 30, 2015

Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern—Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in rth symmetric representation does not change with r, if it is large enough. This fact reflects the non-trivial and previously unknown properties of the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.

DOI: 10.1134/S0021364015120127

1. INTRODUCTION

HOMFLY polynomials are Wilson-loop averages in 3 d Chern—Simons theory [1], which in this simplest model depend only on the topology of the Wilson line (knot). Therefore, one can separate and study the group-theory properties of observables; this is a nontrivial and very interesting problem, for a brief summary of results see [2]. From the quantum field theory perspective knot polynomials are direct generalization of conformal blocks, and this relation [3] provides one of the effective calculation methods in knot theory.

Recent advances in [4, 5], based on the previous considerations in [3—15], provided a way to systematically calculate simplest colored HOMFLY polynomials [16] for a really wide variety of knots including, in particular, the entire Rolfsen table of [17]. This allows us to return to the study of "differential expansions" of [18—27], which was temporarily postponed because of the insufficient "experimental" material.

In this work, we describe empirically obtained properties of these expansions for symmetric representations [r] (where r is the length of the single-line Young diagram). It looks like there are different uni-versality classes of such expansions, characterized by a single integer, which we call "defect" 83£. Moreover, these newly observed properties allow identifying 2(83i + 1) with the power of the Alexander polynomial and lead to a peculiar stability property of symmetrically colored HOMFLY for large enough r: what stabilizes is not the polynomial itself, but the set of its coefficients, i.e., something like the "coordinates" gr j, introduced in [24]. Theoretical analysis of these obser-

vations, proofs and extension to non-(anti)symmetric representations are beyond the scope of the present text.

2. NOTION OF DEFECT

Differential expansion provides a knot-dependent q-deformation (quantization) of the remarkable factorization property [9, 11, 19—21] of colored "special" polynomials at q =1,

JR||

hR(A) = (H1](A))'J = !

V representation R and knot

(!)

which fully defines their dependence on representation (Young diagram) R. Currently these expansions can be well studied only for symmetrically-colored HOMFLY, and we focus on this case in the present paper. The story starts from the fact that

• Hr = H[r] always possesses differential expansion of the following form:

B? (A, q2 ) = !

+

I

s = !

[r] !

G?(A, q){A/q}^{Aq + j}.

s - !

[ s ] ! [ r - s ] !

(2)

j = o

^The article is published in the original.

For generic knot Gs is a non- factorizable Laurent polynomial oL4 and q, but for some knots it can be further factorized. In this formula we use the notation {x} = x — x-1 and quantum number is defined as [n] =

{qn}/{q};

r

what is important, if Gs is divisible by some "dif-

ferential" {Aqk}, the same is true for all other Gy , with s' > s. This property allows one to introduce defect

. • fC fC -f fC

functions vs and = s — 1 — vs :

vf - 1

Gf (A, q) = Ff(A, q) ^ {Aj

j = o

2 f s - 2 - ^s

(3)

= Ff (A, q) n {A4 }

j = o

f . f f . f o

Vs ^ V,. , | ^ Is' for S < S ,

(4)

3. RELATION TO THE ALEXANDER POLYNOMIAL

It is an interesting question, if the value of Sf can also restrict the coefficient functions Ff (A, q).

Immediately observable are two remarkable properties of this kind:

• Gf (A, q) has power 2Sf in q2, i.e., Gf (A, q) =

SL 8* cjqX

f

which are both (!) monotonically increasing function of s,

For example, Sf = 0 whenever Gf is independent of q;

• Alexander polynomial has power 2(Sf + 1) in q2,

i.e., AP\q) = HV =1, q)= Y8* +1 f

¿Si = s-1

i.e., both grow, but not faster than s;

• for A = qNwith any fixed N, positive or negative,

C

qN, q{q}"s ^(qN, q) - {q}s-1, (5)

i.e., the sth term of differential expansion at fixed N is actually of the order {q}2s;

'j = -

f

2J

sf = - Power 2(At )- 1. 2 q

(12)

• it turns out that vs as a function of s has a very special shape, fully parameterized by a single integer Sf > —1, which we call the defect of differential expansion:

defect Sf = -1 ^ |f = s- 2, vf = 1, (6) defect Sf = 0 ^ |f = 0, vf = s- 1, (7)

defect Sf = 1 => |f ~ 2, vf = entier(s---1-J , (8)

f f 2 s f ( s_11

defect S = 2 ^ |s~ -3-, Vs = entier^ --3- J, (9)

f f 3 s f ( s_1J

defect S = 3 => |s ~ —, vs = entier( —4—J, (10)

For Sf ^ 0 these facts are not immediately related:

f

contributing to Alexander polynomials are all Gs with

s < Sf + 1 and they can and do contain much higher powers in q. Moreover, even the product of differentials in the s-term has power in q, which grows quadratically with s and thus with Sf. This means that there are serious cancelations behind the linear law (12).

Since Alexander polynomials are easily available already from [17], the values of v for each knot are easily obtained from this data.

4. TWIST AND TORUS KNOTS For all twist knots the defect is vanishing

8twist = 0. (13)

Instead of torus knots, it is a kind of maximal:

for the 2-strand family S[2' n] = n ^ 3 , for the 3-strand family

(8№ 10124> •••)

o[3, n] 0

S ] = n - 2 ,

for the 4-strand family S

In general

[4, n] = 3 n - 5 = 2 ,

f .. ( s- 1

Vs = entier( Z-^

Sf + 1y Sf + 1

f

f

| s = s 1 V

f

Sf + 1

-s.

(11)

in general

g[m, n] = mn - m - n - 1 (14)

since the power of the Alexander polynomial A/[m, n] is (m - 1)(n - 1).

JETP LETTERS Vol. 101 No. 12 2015

ON THE DEFECT AND STABILITY OF DIFFERENTIAL EXPANSION

833

5. NEGATIVE DEFECT: KTC MUTANTS AND THEIR RELATIVES

Starting from 11 intersections there are cases when the Alexander polynomial is just unity; i.e., the defect

is negative, 8% = -1. According to our general rules,

%

this means that for such knots already G1 is reducible:

% %

G1 ~ {4}. Of course, also all other Gs ~ {A}, because all the terms of the differential expansion are vanishing for A = 1.

This is indeed true for the first example—the celebrated Kinoshita-Terasaka and Conway (KTC) mutants % = 11n42 & 11n34, reconsidered recently in

[5]—and for the next example, available from [17]: % = 12n313 & 12n430. Moreover, the combination of [4] and [5] allows to calculate HOMFLY for KTC mutants for any symmetric representation and validate

(6) in this particular example.

6. SUMMARY: STABILITY AND OTHER PROPERTIES OF DIFFERENTIAL EXPANSION

Take any randomly chosen knot (say, % = 62).

It is easy to observe that, starting from H|2, the sets of coefficients are the same, despite the polynomials are different. At A = q-2 the same happens, beginning

from . Thus, what stabilizes are not the polynomials themselves, but something else, more appropriately associated with the knots. In full accordance with the

vision in [24] this something are the coefficient functions

%

Gs of the differential expansion.

Due to their properties, which are revealed in the present paper, contributing at A = q-N are just the first few terms of the expansion (2):

B? (A = !, q ) = H? (A = !, q)

(!6)

H?(A = !

N

q

(N +1)(s + DrÄr , N

= ! - £ [ N + ! ] G?(A = q )

s = o

s - !

n{q

j = o

{q }s - ! [ s ] !

(!5)

r - N+j

}{q -1},

where the last product is Laurent polynomial in qr and due to (5) the ratio in front of it is an r-independent polynomial. Thus what we get is just a sum of a few polynomials, multiplied by different powers of qr. They do not overlap at large enough r, and this provides an r-independent set of the coefficients, as in the above example.

In fact, one could wish to interpret the remarkable identity [19, 20]

for Alexander polynomials as a manifestation of the same phenomenon at N = 0. However, this is literally so only for 8% = -1 and 8% = 0. Still (16) is true not only for all knots, but actually for all single-hook (and not just single-line) representations R. For such representations (16) is a kind of a dual to (1).

7. CONCLUSIONS

We have studied the "quality" of the differential expansion (2) for symmetrically colored reduced HOMFLY polynomials, which are the typical observables in the simplest possible Yang—Mills theory. If only naive representation-theory properties are taken into account from (1) to restriction l < N on the number l of lines in the Young diagram for particular SL(N), this expansion has the form (2) with irreducible polynomial coefficient functions Gs (A, q). It is well known, however, that sometime Gs are further fac-torized, thus adding more restrictions/structures to the color-dependence of physical observables. Now, when methods were developed to study entire classes of generic knots, we could attack this problem in a systematic way and demonstrate that Gs are always factor-izable for high enough s. The depth of factorization appeared to depend on a single characteristic of the knot, which we originally called defect of the expansion, and further demonstrated that it is linearly related to the degree of Alexander polynomial, what makes it very easy to find.

This factorization universality leads to remarkable kind of stabilization of symmetrically colored HOMFLY, ensuring that increasing r beyond some knot-dependent boundary does not provide new physical (topological) information. This is what one naturally expects, and now we see how this actually works.

Highly desirable is extension of this new insight beyond pure symmetric and antisymmetric representations, but this requires further development of technical tools in conformal, quantum group and ^-matrix theories.

This work was supported in part by the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project no. NSh-1500.2014.2); by the Russian Foundation for Basic Research (project no. 13-02-00478); by the joint grant nos. 15-52-50041-YaF, 14-01-92691-Ind-a, 15-51-52031-NSC-a; by the Brazilian Ministry of Science, Technology and Innovation through the National Counsel of Scientific and Technological Development; and by the Laboratories of Algebraic Geometry and Mathematical Physics, Higher School of Economics.

JETP LETTERS Vol. !0! No. !2 20!5

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JETP LETTERS Vol. 101 No. 12 2015

Приложение Б

Статья 2. Yakov Kononov, Alexei Morozov "Factorization of colored knot polynomials at roots of unity".

Physics Letters B, Volume 747, 30 July 2015, Pages 500-510

Разрешение на копирование: Согласно Соглашению о копирайте автор статьи может использовать полную журнальную версию статьи в своей диссертации при условии, что указан источник.

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Factorization of colored knot polynomials at roots of unity

Ya. Kononovd, A. Morozova'b'c'*

®

CrossMark

a 1TEP, Moscow 117218, Russia

b National Research Nuclear University MEPhI, Moscow 115409, Russia c Institute for Information Transmission Problems, Moscow 127994, Russia d Higher School of Economics, Math Department, Moscow 117312, Russia

a r t i c l e i n f 0

Article history:

Received 22 May 2015

Received in revised form 15 June 2015

Accepted 17 June 2015

Available online 19 June 2015

Editor: M. cvetic

a b s t r a c t

HOMFLY polynomials are the Wilson-loop averages in Chern-Simons theory and depend on four variables: the closed line (knot) in 3d space-time, representation R of the gauge group SU(N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m = 1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: Hr+m = Hr ■ Hm for any A = qN, which is a generalization of the property Hr = H1 for special polynomials at m = 1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2 = e2n 1/|R', turns equal to the special polynomial with A substituted by A|R|, provided R is a singlehook representations (including arbitrary symmetric) - what provides a q — A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots - existence of such universal relations means that these variables are still not unconstrained.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Knot polynomials [1,2] are Wilson loop averages in Chern-Simons theory [3] and their study provides important knowledge and intuition for understanding the properties of gauge-invariant observables in generic Yang-Mills theory. Since Chern-Simons theory is topological, the space-time dependence is completely decoupled and one can extract pure information about the representation (color) dependence. However, the problem of calculating colored HOMFLY polynomials

HR(A, q2) = lrrR pexp^|> A^ (1)

with the gauge group Sl(N) and coupling constant g converted into q2 = exp ^gfjj^ and A = qN, turned to be highly non-trivial. Only recently considerable advances were achieved in [4,5], based on decades of the previous work [6-36], opening a possibility to look for properties, that are valid universally, i.e. for arbitrary knots. In [37] we showed, how this new information leads to immediate breakthrough in the theory of differential expansions [22,28,31]. These expansions provide a non-trivial knot-dependent "quantization" of the archetypical factorization property [21-24]

aR( A) = (aR]( A)) Rl (2)

* Corresponding author at: ITEP, Moscow 117218, Russia.

E-mail addresses: [e-mail] (Ya. Kononov), [e-mail] (A. Morozov).

http://dx.doi.org/10.1016Zj.physletb.2015.06.043

0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

of the special polynomials aR(A) = HR(q2 = 1, A), which are restriction of HOMFLY to q2 = 1, i.e. a kind of their large-N limit. Differential expansion substitutes factorization at q = 1 by expansion at q = 1, which, however, contains finitely many terms with their own pronounced factorization properties. They are best studied for symmetric representation R = [r] :

HK(A, q2|h) = 1 + £

[r]!

s-1

s=1

[s]![r - s]!

hsGf(A, q) •|A/q}f[|Aqr+j}

(3)

j=0

We introduced here an additional parameter h, distinguishing the "level" of differential expansion. HOMFLY polynomial itself arises at h = 1:

HK(A, q2) = HK (A, q2|h = 1) 1. Relations at the roots of unity. Symmetric representations

(4)

In this paper we study another generalization of (2), which preserves its factorized form, but is instead true only at particular values of q - namely, at roots of unity. It turns out that for q2m = 1

H K _ h kk h KK

nr+m n r • n m

q2m=1

(5)

where H'rc = H K] is HOMFLY polynomial in the totally symmetric representation [r] (i.e. Young diagram is a single line of length r).

As an example take the knot K = 62 from the Rolfsen table at m = 2. For q = ±1 we have (2) with a^] = H^ At the other two roots q = ±i we get:

lq=±1

1-2a2+2a4 a4 .

H

62

H

62

H

62

H

62

H

62

H

62

2 A4 + 6 A2 + 3

q=±i A4

_ 2A8 - 2A4 + 1

q=±i

A8

2 A 4 + 6 A 2 + 3 2 A 8 - 2 A 4 + 1

q=±i

q=±i

q=±i

q=±i

A12

2

(2A8 - 2A4 + 1)

A16

(2A4 + 6A2 + 3)(2A8 - 2A4 + 1)

A20

(2A8 - 2A4 + 1)3

A24

also in full accordance with (5).

Original (2) is now a particular case of (5) with m = 1. Since transposition of Young diagram R —> R is equivalent to the substitution

q q-1 [17],

HRc (q2, A) = HR^-, A

the same recursion holds for totally antisymmetric representations [1r] (Young diagram is a column of length r):

ljKK _ rj7C TJJK I

H [1r+m ]= H[1r ]• H [1m] jq2m=!

In fact, (5) is equivalent to a more symmetric statement:

ic

1

' H K _ H K H K >

r+m nr nm) ■ |qgcd(r,m)}

|qr}{q"

|A/q}

' HK HK HK \ : |qr}|qm} IAql

H [1r+m] - H [1rl^ H [1m^ : |qgcd(r,m)} • |Aq}

(6)

(7)

(8)

where gcd(r, m) is the greatest common divisor of r and m, and |x} = x -x-1, so that the quantum number [p] = |qp}/|q} and for coprime r and m the r.h.s. is just [r][m] • |q}|A/q}. The statement is that the r.h.s. factors out from the difference at the l.h.s. at any A and q. It can be interpreted as one more property of the differential expansion (3):

-T//C _"l/K njK _

Hr+m Hr • Hm

r+m /

[m]!

s-1

= y hs • i [r + m]! rf |Aqr+m+j}--rj|Aqr+j} -

l[s]![r + m - s]! 1=0^ ' [s]![r - s]! W1 * } [s]![m - s]! ,=0

j—H|Aqm+j}l •|A/q}•GK +

1

2

5

+ E h

s',s"=l

s'+s"

[r]![m]!

s'-l

s''-1

[s']![s'']![r - s']![m - s'']!

HiAqr+j}f[{ Aqm+j} • {A/q}2 • G^GSK

(9)

j=0

j=0

Many terms in these sums are immediately proportional to the r.h.s. of (8), but not all. Even the factor {q} at m = l is not immediately obvious from (9), but at q = ±1 identity (2) can be additionally used:

(2)

Gs = {A}s-1 • Gl

lq2=1

(10)

Still it turns out - and this is a highly non-trivial additional fact - that proportionality to the r.h.s. of (8) holds independently at each level s = s' + s", i.e. in each order of the h-expansion, thus enhancing (8) to a whole set of quadratic restrictions on the values of GK at roots of unity:

ICrlC

GK = q2 • {A/q} • G£ G'|

lq4=1

H {Aq2r+j} • GK+m = qrm • {A/q} fl {Aqj} • g£g£

... (11)

The nicely-looking relation in the box arises at the order hm+r, but it does not exhaust the set of relations: there are many more, arising at smaller powers of h in between max(r, m) and r + m, but they look less elegant. A useful corollary of (11) is

Gms - qm2s(s-1)/2 • {A/q}s-1 • (Gs)m . q2m - 1

m . 2m

(12)

2. Beyond symmetric representations

To really be a generalization of (2), relations like (5) should hold for arbitrary representations R, not only symmetric. Indeed it looks like there are plenty of them, and they continue with respect to the grading by the level (number of boxes in Young diagram) - all such relations at special values of q are homogeneous in this grading. However it is difficult to find the reliable general rule. In this section we describe the relations at low levels |R| and formulate a plausible general conjecture.

2.1. Extension to [21]

Beyond symmetric representations the story is more complicated, because the analogue of differential expansion (3) is still unknown. Moreover, not much is known about the non-symmetrically HOMFLY at all, even examples are restricted mostly to torus knots. The latest breakthrough in [5] provides answers for rather general knots, but only for R = [21]. Still, this very restricted result allows us to move further.

From the data, obtained on the lines of [5] we conclude empirically that the relevant generalization of (2) for R = [21] is to arbitrary roots of order 6:

L/K _ nK _ nK

H [21] = H [3] = H [111]

q6=1

(H[21] - H[3])

. {q3}{A}

Moreover, the second equality in (13) has its own generalization:

(hK] - H[]) . [r][r - 1]{q}{A}

H[r] = H[1r] |q2r=1 •

H[r] = H

[1r] I q2r-2=1

(13)

(14)

Unfortunately (13) is all what we can check at this moment for rather general knots. In order to move further in non-symmetric case, we need to take a more risky road.

2.2. Implications from torus knots

After the very phenomenon is revealed from analysis of a rather general data, it can be further investigated on a far more restricted data field. Namely, if we believe/assume that there are universal relations between colored HOMFLY at roots of unity, i.e. valid for all knots, their concrete shape can be found by looking at particular knot families. Reliability of such statements is, of course, restricted, and what we get in this way are just conjectures. Still they can be brought to a relatively nice form and it is plausible that they are universally true.

Torus knots provide a natural family to look at, because this is the only one, where colored HOMFLY are available in arbitrary representation. This is because torus knots are more representation-theory than topological objects, and one should be very careful when extending observations made for this family to generic case - still we believe that conjectures below have good chances to be universally reliable.

q

HOMFLY for torus knot [m, n] is given by the Rosso-Jones formula [11,21,17]

HRm,n] = £ cQq2^ XQ/XR, (15)

IQ |=m|R|

where k q is the content of the diagram, and cQ are matrix elements of the so-called Adams endomorphism in the basis of Schur polynomials.

2.3. Conjecture

From the study of torus knots we make a conjecture, which is presumably valid for all knots:

(16)

iiK ? UK iiK HR+M — Hm • HR

V connected skew diagram M of width one with |M|= m

q2m=1

provided both R and R + M are Young diagrams. The following picture is an explanation of what we mean by R + M:

connected M

of unit width