Quantum R-matrices, classical integrability and knot invariants - - PowerPoint PPT Presentation

Quantum R-matrices, classical integrability and knot invariants

Andrei Mironov

P.N.Lebedev Physics Institute and ITEP

Integrability and Combinatorics, 2014

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 2 / 25

Basic examples A generating function of the non-linear Schr¨

• dinger equation (at infinite coupling) −

→ classical non-linear Schr¨

• dinger at finite

coupling Master T-operator for inhomogeneous XXX GL(N) spin chain with twisted boundary conditions − → Hirota blinear equation for mKP Partition function of the 6-vertex model with inhomogeneities (determinant formula) − → KP τ-function in Miwa variables . . . Refs A.Its, A.Izergin, V.Korepin, N.Slavnov (1990) A.Zabrodin (A.Alexandrov, V.Kazakov, S.Leurent, Z.Tsuboi) A.Izegin (1987) M.Jimbo, T.Miwa, M.Sato; T.T.Wu, B.M.McCoy, C.A.Tracy, E.Barouch; D.Bernard, A.LeClair; ... Goal for knot theory: from knots to equations Yet another approach: quantum A-polynomials (S.Garoufalidis, S.Gukov etc)

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 3 / 25

Plan

1

HOMFLY-PT, generating functions and τ-functions

2

Rosso-Jones formula for torus knots/links

3

Tau-functions for torus knots

4

HOMFLY-PT via R-matrices

5

Deformations Non-torus knots

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 4 / 25

HOMFLY-PT, generating functions and τ-functions

HOMFLY-PT HK

R(q|A)

• A=qN =
• TrRP exp
• K

A

• SU(N)

3d Chern-Simons theory with the gauge group SU(N) and the action S = κ 4π

• d3x

Tr

• AdA + 2

3A3 q = exp 2πi κ + N Skein relations for HK

(q|A):

HK

R(q|A) is proportional to a Laurent polynomial in q and A

A

• ✒

❅ ❅ ❅ ❅ ■ − A−1 ❅ ❅ ■ ❅ ❅

• ✒

= {q} ✻✻ {x} ≡ x − 1 x

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 5 / 25

Color HOMFLY-PT: cabling Trefoil

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 6 / 25

Trefoil: simplest torus knot

Examples H[2,3]

[1]

=

• (q2 + q−2)A − A−1 {A}

{q} H[2,3]

[2]

=

• (q12 + q6 + q4 + 1)A2 − q8 − q6 − q2 − 1 + q2A−2

q4 {A}{Aq} {q}{q2} H[2,3]

[11] =

• (q−12 + q−6 + q−4 + 1)A2 − q−8 − q−6 − q−2 − 1 + q−2A−2 {A}{A/q}

q4{q}{q2}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 7 / 25

Ooguri-Vafa partition function as a τ-function

ZK(q, A) =

• R

HK

R(q, A)SR(p) =

• exp
• n

1 nTr

• K

Adx n TrV n

• pk = TrV k =

i vk i are external sources, SR(p) are Schur functions, Tr is taken over the fundamental

representation. Ooguri-Vafa conjecture ZK(q, A) describes the topological string on the resolved conifold. When is a τ-function? Unknot: Hopf link:

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25

Ooguri-Vafa partition function as a τ-function

ZK(q, A) =

• R

HK

R(q, A)SR(p) =

• exp
• n

1 nTr

• K

Adx n TrV n

• pk = TrV k =

i vk i are external sources, SR(p) are Schur functions, Tr is taken over the fundamental

representation. Ooguri-Vafa conjecture ZK(q, A) describes the topological string on the resolved conifold. When is a τ-function? Unknot: Hopf link:

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25

Ooguri-Vafa partition function as a τ-function

ZK(q, A) =

• R

HK

R(q, A)SR(p) =

• exp
• n

1 nTr

• K

Adx n TrV n

• pk = TrV k =

i vk i are external sources, SR(p) are Schur functions, Tr is taken over the fundamental

representation. Ooguri-Vafa conjecture ZK(q, A) describes the topological string on the resolved conifold. When is a τ-function? Unknot: Hopf link:

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25

Ooguri-Vafa partition function as a τ-function

ZK(q, A) =

• R

HK

R(q, A)SR(p) =

• exp
• n

1 nTr

• K

Adx n TrV n

• pk = TrV k =

i vk i are external sources, SR(p) are Schur functions, Tr is taken over the fundamental

representation. Ooguri-Vafa conjecture ZK(q, A) describes the topological string on the resolved conifold. When is a τ-function? Unknot: Hopf link:

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25

τ-function as sum over characters

τ(t) =

• R

gRSR(p) tk = pk/k What are the conditions for gR? g[22]g[0] − g[21]g[1] + g[2]g[11] = 0 g[32]g[0] − g[31]g[1] + g[3]g[11] = 0 g[221]g[0] − g[211]g[1] + g[2]g[111] = 0 . . . These are the Pl¨ ucker relations. Their general solution is: cR = det

ij MRi−i,j

A particular example

• R

SR(¯ p)SR(p) exp

k

ξkCk(R)

• Ck =
• i
• (Ri − i + 1/2)k − (−i + 1/2)k

is a τ-function of both the KP and Toda lattice hierarchy.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25

τ-function as sum over characters

τ(t) =

• R

gRSR(p) tk = pk/k What are the conditions for gR? g[22]g[0] − g[21]g[1] + g[2]g[11] = 0 g[32]g[0] − g[31]g[1] + g[3]g[11] = 0 g[221]g[0] − g[211]g[1] + g[2]g[111] = 0 . . . These are the Pl¨ ucker relations. Their general solution is: cR = det

ij MRi−i,j

A particular example

• R

SR(¯ p)SR(p) exp

k

ξkCk(R)

• Ck =
• i
• (Ri − i + 1/2)k − (−i + 1/2)k

is a τ-function of both the KP and Toda lattice hierarchy.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25

τ-function as sum over characters

τ(t) =

• R

gRSR(p) tk = pk/k What are the conditions for gR? g[22]g[0] − g[21]g[1] + g[2]g[11] = 0 g[32]g[0] − g[31]g[1] + g[3]g[11] = 0 g[221]g[0] − g[211]g[1] + g[2]g[111] = 0 . . . These are the Pl¨ ucker relations. Their general solution is: cR = det

ij MRi−i,j

A particular example

• R

SR(¯ p)SR(p) exp

k

ξkCk(R)

• Ck =
• i
• (Ri − i + 1/2)k − (−i + 1/2)k

is a τ-function of both the KP and Toda lattice hierarchy.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25

τ-function as sum over characters

τ(t) =

• R

gRSR(p) tk = pk/k What are the conditions for gR? g[22]g[0] − g[21]g[1] + g[2]g[11] = 0 g[32]g[0] − g[31]g[1] + g[3]g[11] = 0 g[221]g[0] − g[211]g[1] + g[2]g[111] = 0 . . . These are the Pl¨ ucker relations. Their general solution is: cR = det

ij MRi−i,j

A particular example

• R

SR(¯ p)SR(p) exp

k

ξkCk(R)

• Ck =
• i
• (Ri − i + 1/2)k − (−i + 1/2)k

is a τ-function of both the KP and Toda lattice hierarchy.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25

Torus knots/links and braid representation

Closed braid: If m and n are mutually prime, it is torus. Otherwise, it is a link with l components if l is the largest common divisor of m and n.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 10 / 25

Adams operation. Pletism

Adams operation: step 1 p[m]

k

= pmk vi − → vm

i

Adams operation: step 2 For knots: SR(p[m]) =

• Q⊢m|R|

CQ

RSQ(p)

For links with l components:

l

• a=1

SRa(p[m]) =

• Q⊢m|R|

CQ

R1...RlSQ(p)

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25

Adams operation. Pletism

Adams operation: step 1 p[m]

k

= pmk vi − → vm

i

Adams operation: step 2 For knots: SR(p[m]) =

• Q⊢m|R|

CQ

RSQ(p)

For links with l components:

l

• a=1

SRa(p[m]) =

• Q⊢m|R|

CQ

R1...RlSQ(p)

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25

Adams operation. Pletism

Adams operation: step 1 p[m]

k

= pmk vi − → vm

i

Adams operation: step 2 For knots: SR(p[m]) =

• Q⊢m|R|

CQ

RSQ(p)

For links with l components:

l

• a=1

SRa(p[m]) =

• Q⊢m|R|

CQ

R1...RlSQ(p)

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25

Rosso-Jones formula

For knots: Hm,n

R

(q|A) =

• Q⊢m|R|

q

n 2m C2(Q)CQ

RS∗ Q

Here S∗

Q ≡ SQ(p∗)

pk = {Ak} {qk} For links with l components: Hm,n

R1...Rl(q|A) =

• Q⊢m|R|

q

n m C2(Q)CQ

R1...RlS∗ Q

S∗

Q is the HOMFLY-PT for unknot and is equal to the quantum dimension of Q:

S∗

Q =

• (i,j)∈Q

{Aqi−j} {qhi,j}

A=qN

− →

• (i,j)∈Q

[N + i − j]q [hi,j]q hi,j is the hook length.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 12 / 25

Rosso-Jones formula

For knots: Hm,n

R

(q|A) =

• Q⊢m|R|

q

n 2m C2(Q)CQ

RS∗ Q

Here S∗

Q ≡ SQ(p∗)

pk = {Ak} {qk} For links with l components: Hm,n

R1...Rl(q|A) =

• Q⊢m|R|

q

n m C2(Q)CQ

R1...RlS∗ Q

S∗

Q is the HOMFLY-PT for unknot and is equal to the quantum dimension of Q:

S∗

Q =

• (i,j)∈Q

{Aqi−j} {qhi,j}

A=qN

− →

• (i,j)∈Q

[N + i − j]q [hi,j]q hi,j is the hook length.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 12 / 25

Rosso-Jones formula

For knots: Hm,n

R

(q|A) =

• Q⊢m|R|

q

n 2m C2(Q)CQ

RS∗ Q

Here S∗

Q ≡ SQ(p∗)

pk = {Ak} {qk} For links with l components: Hm,n

R1...Rl(q|A) =

• Q⊢m|R|

q

n m C2(Q)CQ

R1...RlS∗ Q

S∗

Q is the HOMFLY-PT for unknot and is equal to the quantum dimension of Q:

S∗

Q =

• (i,j)∈Q

{Aqi−j} {qhi,j}

A=qN

− →

• (i,j)∈Q

[N + i − j]q [hi,j]q hi,j is the hook length.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 12 / 25

Example of 2-strand knot/link ✱ ✱ m = 2, odd n corresponds to a knot. C[2] = 2, C[11] = −2 and CQ

R:

S[1](p(2)) = p2 = p2

1

2 + p2 2

• −

p2

1

2 − p2 2

• = S[2](p) − S[11](p),

H2,2k+1

[1]

= q2k+1S∗

[2] − q−2k−1S∗ [1,1]

Similarly, even n corresponds to a link: S[1](p)2 = p2

1 =

p2

1

2 + p2 2

• +

p2

1

2 − p2 2

• = S[2](p) + S[11](p)

H2,2k

[1],[1] = q2kS∗ [2] + q−2kS∗ [1,1]

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 13 / 25

Topological invariance on the topological locus

H2,3

[1] (p) = q3S[2](p) − q−3S[11](p)

⇓

pk= Ak−A−k qk−q−k

H2,3

[1] (q|A) =

• (q2 + q−2)A − A−1 {A}

{q} H3,2

[1] (p) = q4S[3](p) − S[21](p) + q−4S[111](p)

⇓

pk= Ak−A−k qk−q−k

H3,2

[1] (q|A) = A ×

• (q2 + q−2)A − A−1 {A}

{q}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 14 / 25

Topological invariance on the topological locus

H2,3

[1] (p) = q3S[2](p) − q−3S[11](p)

⇓

pk= Ak−A−k qk−q−k

H2,3

[1] (q|A) =

• (q2 + q−2)A − A−1 {A}

{q} H3,2

[1] (p) = q4S[3](p) − S[21](p) + q−4S[111](p)

⇓

pk= Ak−A−k qk−q−k

H3,2

[1] (q|A) = A ×

• (q2 + q−2)A − A−1 {A}

{q}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 14 / 25

Topological invariance on the topological locus

H2,3

[1] (p) = q3S[2](p) − q−3S[11](p)

⇓

pk= Ak−A−k qk−q−k

H2,3

[1] (q|A) =

• (q2 + q−2)A − A−1 {A}

{q} H3,2

[1] (p) = q4S[3](p) − S[21](p) + q−4S[111](p)

⇓

pk= Ak−A−k qk−q−k

H3,2

[1] (q|A) = A ×

• (q2 + q−2)A − A−1 {A}

{q}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 14 / 25

Topological invariance on the topological locus

H2,3

[1] (p) = q3S[2](p) − q−3S[11](p)

⇓

pk= Ak−A−k qk−q−k

H2,3

[1] (q|A) =

• (q2 + q−2)A − A−1 {A}

{q} H3,2

[1] (p) = q4S[3](p) − S[21](p) + q−4S[111](p)

⇓

pk= Ak−A−k qk−q−k

H3,2

[1] (q|A) = A ×

• (q2 + q−2)A − A−1 {A}

{q}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 14 / 25

Topological invariance on the topological locus

H2,3

[1] (p) = q3S[2](p) − q−3S[11](p)

⇓

pk= Ak−A−k qk−q−k

H2,3

[1] (q|A) =

• (q2 + q−2)A − A−1 {A}

{q} H3,2

[1] (p) = q4S[3](p) − S[21](p) + q−4S[111](p)

⇓

pk= Ak−A−k qk−q−k

H3,2

[1] (q|A) = A ×

• (q2 + q−2)A − A−1 {A}

{q}

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 14 / 25

Cut-and-join operator

Simplest cut-and-join operator ˆ W[2]: ˆ W[2]SQ(t) = C2(Q)SQ(t) Manifestly: ˆ W[2] =

• a,b
• abtatb

∂ ∂ta+b + (a + b)ta+b ∂2 ∂ta∂tb

• The HOMFLY-PT for torus knots without fixing pk = p∗

k:

Z(t, ¯ t) =

• R

SR(¯ t)Hm,n

R

(t, ¯ t) =

• R

SR(¯ t)

• Q⊢m|R|

q

n m C2(Q)CQ

RSQ(t) =

=

• R

SR(¯ t)q− n

m ˆ

W (t)

• Q⊢m|R|

CQ

RS∗ Q = q− n

m ˆ

W (t) R

SR(t[m])SR(¯ t) From the Cauchy formula

• R

SR(t)SR(¯ t) = exp

• k

ktk¯ tk Finally Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk¯

tk

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 15 / 25

Cut-and-join operator

Simplest cut-and-join operator ˆ W[2]: ˆ W[2]SQ(t) = C2(Q)SQ(t) Manifestly: ˆ W[2] =

• a,b
• abtatb

∂ ∂ta+b + (a + b)ta+b ∂2 ∂ta∂tb

• The HOMFLY-PT for torus knots without fixing pk = p∗

k:

Z(t, ¯ t) =

• R

SR(¯ t)Hm,n

R

(t, ¯ t) =

• R

SR(¯ t)

• Q⊢m|R|

q

n m C2(Q)CQ

RSQ(t) =

=

• R

SR(¯ t)q− n

m ˆ

W (t)

• Q⊢m|R|

CQ

RS∗ Q = q− n

m ˆ

W (t) R

SR(t[m])SR(¯ t) From the Cauchy formula

• R

SR(t)SR(¯ t) = exp

• k

ktk¯ tk Finally Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk¯

tk

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 15 / 25

Ooguri-Vafa partition function for the torus knot as a τ-function

Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk¯

tk

Z(t,¯ t) is not a topological invariant, it is a braid invariant a τ-function of KP hierarchy w.r.t. times tk, since: (i) the exponential of t-variables is the simplest KP τ-function; (ii) the cut-and-join operator ˆ W is an element of the group GL(∞), its action preserves KP-integrability in t. Another representation: Z(t, ¯ t) =

• R

SR(¯ t)SR(p) exp 1 2C2(R)

• with ¯

tk = 0 unless k = 0 mod m. For torus links, it is still a τ-function: Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk

l

a=1 ¯

t(a)

k

• A.Mironov

(LPI/ITEP) Knots and tau-functions 2014 16 / 25

Ooguri-Vafa partition function for the torus knot as a τ-function

Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk¯

tk

Z(t,¯ t) is not a topological invariant, it is a braid invariant a τ-function of KP hierarchy w.r.t. times tk, since: (i) the exponential of t-variables is the simplest KP τ-function; (ii) the cut-and-join operator ˆ W is an element of the group GL(∞), its action preserves KP-integrability in t. Another representation: Z(t, ¯ t) =

• R

SR(¯ t)SR(p) exp 1 2C2(R)

• with ¯

tk = 0 unless k = 0 mod m. For torus links, it is still a τ-function: Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk

l

a=1 ¯

t(a)

k

• A.Mironov

(LPI/ITEP) Knots and tau-functions 2014 16 / 25

Ooguri-Vafa partition function for the torus knot as a τ-function

Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk¯

tk

Z(t,¯ t) is not a topological invariant, it is a braid invariant a τ-function of KP hierarchy w.r.t. times tk, since: (i) the exponential of t-variables is the simplest KP τ-function; (ii) the cut-and-join operator ˆ W is an element of the group GL(∞), its action preserves KP-integrability in t. Another representation: Z(t, ¯ t) =

• R

SR(¯ t)SR(p) exp 1 2C2(R)

• with ¯

tk = 0 unless k = 0 mod m. For torus links, it is still a τ-function: Z(t, ¯ t) = q− n

m ˆ

W (t)e

• k mktmk

l

a=1 ¯

t(a)

k

• A.Mironov

(LPI/ITEP) Knots and tau-functions 2014 16 / 25

HOMFLY-PT via R-matrices Redemeister move

First Redemeister move. Second Redemeister move. Third Redemeister move.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 17 / 25

Reshetikhin-Turaev formula

Example: 3-strand closed braid

• ✪

✪ ✪ HB

R = "tr"R⊗m

• B
• B = R2

23R−2 12 R23R−3 12

HOMFLY-PT HK

R(q|A = qN): representation R of SUq(N), ρ is a sum of positive roots:

HB

R = trR⊗m

• (qρ)⊗mB
• A.Mironov

(LPI/ITEP) Knots and tau-functions 2014 18 / 25

Reshetikhin-Turaev formula

Example: 3-strand closed braid

• ✪

✪ ✪ HB

R = "tr"R⊗m

• B
• B = R2

23R−2 12 R23R−3 12

HOMFLY-PT HK

R(q|A = qN): representation R of SUq(N), ρ is a sum of positive roots:

HB

R = trR⊗m

• (qρ)⊗mB
• A.Mironov

(LPI/ITEP) Knots and tau-functions 2014 18 / 25

R-matrix

= The third Redemeister move R12R23R12 = R23R12R23 The Yang-Baxter equation R12R13R23 = R23R13R12 R is the standard R-matrix of SUq(N) multipled by the permutation operator. Then, R∆(g)R−1 = ∆(g) Hence R is diagonal on the irreducible representation R with eigenvalue rR ∼ ±q

1 2 C2(R) A.Mironov (LPI/ITEP) Knots and tau-functions 2014 19 / 25

R-matrix

= The third Redemeister move R12R23R12 = R23R12R23 The Yang-Baxter equation R12R13R23 = R23R13R12 R is the standard R-matrix of SUq(N) multipled by the permutation operator. Then, R∆(g)R−1 = ∆(g) Hence R is diagonal on the irreducible representation R with eigenvalue rR ∼ ±q

1 2 C2(R) A.Mironov (LPI/ITEP) Knots and tau-functions 2014 19 / 25

Basis of irreducible representations

Example of 2-strand knot/link ✱ ✱ [1] × [1] = [2] + [11] Eigenvalues: r[2] = q, r[11] = − 1

q

H2,2k

[1]

= tr[1]

• qρ⊗2R2k

= q2kS∗

[2]+q−2kS∗ [1,1],

H2,2k+1

[1]

= tr[1]

• qρ⊗2R2k+1

= q2k+1S∗

[2]−q−2k−1S∗ [1,1]

Compare with the Rosso-Jones formula.

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 20 / 25

Basis of irreducible representations

HB

R = trR⊗m

• (qρ)⊗mB
• =
• Q⊢m|R|

hQ

R[B] S∗ Q(A)

where S∗

Q(q|A) = trQ∈R⊗m(qρ)⊗m

Coefficients hQ

R[B] do not depend on A, i.e. on N, thus, they can be evaluated from analysis of

arbitrary group SUq(N). Instead, these coefficients can be represented as traces in auxiliary spaces of intertwiner operators MQ

Rm, whose dimension is the number dimMQ Rm = N Q Rm of times the irreducible representation Q

appears in the m-th tensor power of the representation R, R⊗m =

• Q⊢m|R|

MQ

Rm ⊗ Q

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 21 / 25

Example: 3-strand braid calculation [1]3 = [3] + 2 [21] + [111] H[1](q|A) = tr

• R ⊗ I

a1 I ⊗ R b1 R ⊗ I a2 I ⊗ R b2 . . .

• =
• Q=[111],[21],[3]

Tr

• ˆ

RQ

12

a1 ˆ RQ

23

b1 ˆ RQ

12

a2 ˆ RQ

23

b2 . . .

• = qa1+b1+a2+b2+...S∗

3 +

• −1

q a1+b1+a2+b2+... S∗

111+

+Tr    q − 1

q

a1 C S −S C q − 1

q

b1 C −S S C

• ×

× q − 1

q

a2 C S −S C q − 1

q

b2 C −S S C

• . . .

   S∗

21

C = 1 [2]q , S =

• [3]q

[2]q

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 22 / 25

Example: 3-strand braid calculation [1]3 = [3] + 2 [21] + [111] H[1](q|A) = tr

• R ⊗ I

a1 I ⊗ R b1 R ⊗ I a2 I ⊗ R b2 . . .

• =
• Q=[111],[21],[3]

Tr

• ˆ

RQ

12

a1 ˆ RQ

23

b1 ˆ RQ

12

a2 ˆ RQ

23

b2 . . .

• = qa1+b1+a2+b2+...S∗

3 +

• −1

q a1+b1+a2+b2+... S∗

111+

+Tr    q − 1

q

a1 C S −S C q − 1

q

b1 C −S S C

• ×

× q − 1

q

a2 C S −S C q − 1

q

b2 C −S S C

• . . .

   S∗

21

C = 1 [2]q , S =

• [3]q

[2]q

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 22 / 25

Example: 3-strand braid calculation [1]3 = [3] + 2 [21] + [111] H[1](q|A) = tr

• R ⊗ I

a1 I ⊗ R b1 R ⊗ I a2 I ⊗ R b2 . . .

• =
• Q=[111],[21],[3]

Tr

• ˆ

RQ

12

a1 ˆ RQ

23

b1 ˆ RQ

12

a2 ˆ RQ

23

b2 . . .

• = qa1+b1+a2+b2+...S∗

3 +

• −1

q a1+b1+a2+b2+... S∗

111+

+Tr    q − 1

q

a1 C S −S C q − 1

q

b1 C −S S C

• ×

× q − 1

q

a2 C S −S C q − 1

q

b2 C −S S C

• . . .

   S∗

21

C = 1 [2]q , S =

• [3]q

[2]q

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 22 / 25

Integrability for non-torus knots

Braid invariants and off-shell HOMFLY-PT Braid invariant: HB

R(p) =

• Q⊢m|R|

hQ

R[B] SQ(p)

Do the Pl¨ ucker relations satisfy? The first Pl¨ ucker relation is g[22]g[0] − g[21]g[1] + g[2]g[11] = 0

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 23 / 25

Integrability for non-torus knots

Braid invariants and off-shell HOMFLY-PT Braid invariant: HB

R(p) =

• Q⊢m|R|

hQ

R[B] SQ(p)

Do the Pl¨ ucker relations satisfy? The first Pl¨ ucker relation is g[22]g[0] − g[21]g[1] + g[2]g[11] = 0

A.Mironov (LPI/ITEP) Knots and tau-functions 2014 23 / 25

4-strand example In the case of non-toric knots the Ooguri-Vafa partition function is typically not a KP τ-function. Consider the first non-trivial Pl¨ ucker relation for gQ =

R hQ RSR(¯

t) and a 4-strand knot. Since g0 = 1, g[1] = g[2] = g[11] = g[21] = 0 in this case, one inevitably should have g[22] = h[22]

[1] = 0. This is the case

for the torus knots, and not typically the case for others. For the first 4-strand knots from the Rolfsen table (up to 8 crossings): knot h[22]

[1]

61 q−1 − q1 72 −q7 + q5 − 2q3 + 3q1 − 3q−1 + 2q−3 − q−5 + q−7 74 (q − q−1)(q6 − q4 + 3q2 − 1 + 3q−2 − q−4 + q−6) 76 −q7 + 2q5 − 3q3 + 3q1 − 3q−1 + 3q−3 − 2q−5 + q−7 77 −q7 + 3q5 − 4q3 + 5q1 − 5q−1 + 4q−3 − 3q−5 + q−7 84 (q − q−1)(q4 − q2 + 1 − q−2 + q−4) 86 (q − q−1)(q2 + 1 + q−2)(q2 − 1 + q−2) 811 −q3 + q−3 813 (q − q−1)(q4 − q2 + 1 − q−2 + q−4) 814 (q − q−1)(q2 + 1 + q−2)(q2 − 1 + q−2) 815 (q − q−1)(q6 − 2q4 + 2q2 − 3 + 2q−2 − 2q−4 + q−6) Thus, for all these knots the Pl¨ ucker relation is not satisfied (torus knots with 4 strands have more than 8 crossings).

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Thank you for your attention!

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