Advances in Knot Polynomials 21 October 2016 Advances in Knot - - PowerPoint PPT Presentation

Advances in Knot Polynomials

21 October 2016

Advances in Knot Polynomials 21 October 2016 1 / 49

ABSTRACT Review of achievements and problems in the theory of colored knot

• polynomials. Accent is on the current mystery around the differential

expansion and Racah matrices (6j-symbols) in rectangular representations.

Advances in Knot Polynomials 21 October 2016 2 / 49

J.W.Alexander, Trans.Amer.Math.Soc. 30 (2) (1928) 275-306 V.F.R.Jones, Invent.Math. 72 (1983) 1 P.Freyd, D.Yetter, J.Hoste, W.B.R.Lickorish, K.Millet, A.Ocneanu,

• Bull. AMS. 12 (1985) 239

J.H.Przytycki and K.P.Traczyk, Kobe J. Math. 4 (1987) 115-139 L.Kauffman,Topology 26 (1987) 395

Advances in Knot Polynomials 21 October 2016 3 / 49

Colored HOMFLY polynomial: HK

R(A, q) =

• TrR P exp
• K

A

• average with CS action with the gauge group G = SL(N):

κ

• M

Tr

• AdA + 2

3A3

• q = exp

2πi κ + N A = qN

Advances in Knot Polynomials 21 October 2016 4 / 49

HOMFLY are exactly-calculable non-perturbative averages in gauge QFT

Advances in Knot Polynomials 21 October 2016 5 / 49

For simply-connected M (R3 or S3): Calculation of HOMFLY Properties of HOMFLY Generalizations of HOMFLY Relations to other theories Other M HOMFLY for virtual knots

Advances in Knot Polynomials 21 October 2016 6 / 49

A.Mironov A.Anokhina, S.Arthamonov, V.Dolotin, P.Dunin-Barkovski, D.Galakhov, H.Itoyama, Ya.Kononov, D.Melnikov, An.Morozov, P.Ramadevi, Vivek Singh, Sh.Shakirov, A.Sleptsov, A.Smirnov

Advances in Knot Polynomials 21 October 2016 7 / 49

Calculation of HOMFLY

Modernized Reshetikhin-Turaev calculus Paths in representation graphs Eigenvalue hypothesis Arborescent knots – new effective theory Fingered braids – the most efficient tool at the moment Evolution/family method . . .

Advances in Knot Polynomials 21 October 2016 8 / 49

❅ ❅ ■ ✻

• ✒

CS

❄

projection to 2d lattice theory on arbitrary graphs

• ✠

❅ ❅ ❘

algebra geometry RT KhR hypercube

• ther 2d

Seifert surfaces CFT 3d

✟✟✟✟ ✯

higher d

Advances in Knot Polynomials 21 October 2016 9 / 49

✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

R±1

12

R±1

23

R±1

34

i j k l R R R R tr qρ . . .

• Q DQ · TrWQ . . .

R ⊗ R

⊕ Y

⊗R ⊗ R = ⊕ Q ⊗ WQ RQ

23 = UQ 23RQ 12UQ 23 †

RQ

12 = diag

• ǫY · qκY
• RT calculus

traditional modern

Advances in Knot Polynomials 21 October 2016 10 / 49

Fundamental representation R = : skein relations R − R−1 = q − q−1 paths in representation tree

Advances in Knot Polynomials 21 October 2016 11 / 49

Other representations R: cabling method eigenvalue hypothesis tree calculus – under construction not just braids

Advances in Knot Polynomials 21 October 2016 12 / 49

BRAID CALCULUS Needed is entire collection of mixing matrices For 3 strands needed are only Racah matrices S(Q)

Y ′Y ′′

• (R ⊗ R

Y ′

) ⊗ R − → Q

• −

→

• R ⊗ (R ⊗ R

Y ′′

) − → Q

• but "inclusive": for all Q ∈ R⊗3

This is realistic, but too few knots are 3-strand

Advances in Knot Polynomials 21 October 2016 13 / 49

ARBORESCENT KNOTS/LINKS made from fingers, propagators and vertices

Advances in Knot Polynomials 21 October 2016 14 / 49

ARBORESCENT (double-fat) KNOTS/LINKS made from fingers, propagators and vertices

Advances in Knot Polynomials 21 October 2016 15 / 49

Advances in Knot Polynomials 21 October 2016 16 / 49

Needed are just two "exclusive" Racah matrices ¯ S :

• (R ⊗ ¯

R) ⊗ R − → R

• −

→

• R ⊗ (¯

R ⊗ R) − → R

• and

S :

• (¯

R ⊗ R) ⊗ R − → R

• −

→

• ¯

R ⊗ (R ⊗ R) − → R

• Advances in Knot Polynomials

21 October 2016 17 / 49

¯ T ¯ S ¯ T = ST −1S† T and ¯ T are diagonal matrices

Advances in Knot Polynomials 21 October 2016 18 / 49

FINGERED BRAIDS

✲ ✲

n1

✲ ✲

n2

✲ ✲

n3

✲ ✲

n4

✲ ✲

n5

✲ ✲ ✲

n6

✲ ✛

n7

✲ ✲ ✲

Advances in Knot Polynomials 21 October 2016 19 / 49

FINGERED BRAIDS allow to handle more complicated knots by using less strands Three-strand fingered braids are already quite rich but even for them calculus is still hard

Advances in Knot Polynomials 21 October 2016 20 / 49

side stories: The structure of the space of knots Effective gauge field theory for arborescent knots Gauge invariance, vertices and loops Rectangular and non-rectangular representations

Advances in Knot Polynomials 21 October 2016 21 / 49

States:

✻ ✻ ❄ ❄

σ

✻ ❄ ✻ ❄

ϕ

✻ ❄❄ ✻

φ and conjugates:

✻ ✻ ❄ ❄

σ∗

✻ ❄ ✻ ❄

ϕ∗

✻ ❄ ❄ ✻

φ∗ Each of them carries indices σAB − → σXαβ with the gauge group acting by two orthogonal matrices A and B: σX,α,β − →

• α′β′

Aαα′Bββ′σX,α′β′

Advances in Knot Polynomials 21 October 2016 22 / 49

Quadratic terms in the Lagrangian are: "local"ones σXT n

XσX = σX,αβT n X,αα′σX,α′β

ϕX ¯ T 2n

X ϕX,

φX ¯ T 2n

X φX,

ϕX ¯ T 2n−1

X

φX, φX ¯ T 2n

X ϕX

plus conjugates, "non-local"ones σ∗

XS† XY φY ,

φ∗

XSXY σY ,

ϕ∗

X ¯

SXY ϕY (note that there are no terms φ∗

XφY )

Advances in Knot Polynomials 21 October 2016 23 / 49

Topologically allowed vertices are Γ(1) ∼ σ3

X,

Γ(2) ∼ ϕ3

X,

Γ(3) ∼ φ2

XϕX

The problem is, however, to deal with the Greek indices in Γα,β,γΦα,βΦβ,γΦγ,α. A naive anzatz like tr σ3

X with the trace in Greek

indices would be good for a transformation law σ − → AσA†, but it violates σ − → AσB with independent A and B. This means that at the representational level one can not get a gauge invariant description of our knot polynomials. If one calculates the Feynman diagram for some particular choice of S (in a particular gauge), the answer differs in other gauges so that there should be some "handy"compensational rule attached to the answer.

Advances in Knot Polynomials 21 October 2016 24 / 49

. . .

❆ ❆ ✁ ✁ ✁ ✁ ❆ ❆

. . .

❍✟ ✟❍ ✟ ✟ ❍❍ ✯ ❥ ❑ ☛ ✕ ❯

2n 2m

Advances in Knot Polynomials 21 October 2016 25 / 49

EVOLUTION for twist and double braid knots

H(m,n)

R

=

• µ,ν∈R⊗¯

R

• DµDν

DR ¯ Sµν Λ2m

µ Λ2n ν

Advances in Knot Polynomials 21 October 2016 26 / 49

Properties of HOMFLY

Polynomiality and integralities Factorizations Equations Hurwitz integrability Vogel’s universality (unification of E8-sectors of all Lie algebras) Differential expansions . . .

Advances in Knot Polynomials 21 October 2016 27 / 49

. . .

Advances in Knot Polynomials 21 October 2016 28 / 49

DIFFERENTIAL EXPANSION

Advances in Knot Polynomials 21 October 2016 29 / 49

H41

[1] = A2 − q2 + 1 − q−2 + A−2

= 1 + {Aq}{A/q} A = qN {x} = x − 1/x [n] = qn − q−n q − q−1 = {qn} {q} Differentials {Aqn} ∼ [N + n]

Advances in Knot Polynomials 21 October 2016 30 / 49

H41

[1] = 1 + {Aq}{A/q}

H41

[2] = 1 + [2]{Aq2}{A/q} + {Aq3}{Aq2}{A}{A/q}

H41

[3] = 1 + [3]{Aq3}{A/q} + [3]{Aq4}{Aq3}{A}{A/q}+

+{Aq5}{Aq4}{Aq3}{Aq}{A}{A/q} . . .

Advances in Knot Polynomials 21 October 2016 31 / 49

Differential expansion

Equations . . . Superpolynomials տ ↑ ր H41

[1] = 1 + {Aq}{A/q}

H41

[2] = 1 + [2]{Aq2}{A/q} + {Aq3}{Aq2}{A}{A/q}

ւ ց Other representations Other knots H41

[rs] = λ∈[rs] D˜ λ(r)Dλ(s)Z λ r|s

HK

[1] = 1 + G K [1](q, A){Aq}{A/q}

Advances in Knot Polynomials 21 October 2016 32 / 49

Z λ

r|s(A, q) =

• ∈λ

{Aqr+a′()−l′()}{Aq−s+a′()−l′()}

✛ ✻ ✲ ❄

a′ l′ a l Dλ(N) =

• ∈λ

{Aq−l′()+a′()} {qa()+l()+1} =

• ∈λ

[N − l′() + a′()] [a() + l() + 1]

Advances in Knot Polynomials 21 October 2016 33 / 49

Other knots: HK

[1] = 1 + G K [1](q, A){Aq}{A/q}

HK

[2] = 1 + [2]G K [1](q, A){Aq2}{A/q} + G K [2](q, A){Aq2}{A}{A/q}

• nly for defect zero:

HK

[2] = 1 + [2]F K [1](q, A){Aq2}{A/q} + F K0 [2] (q, A){Aq3}{Aq2}{A}{A/q}

defect = powerq2

• AlK

[1]

• 1

Advances in Knot Polynomials 21 October 2016 34 / 49

Double braids have defect zero and very special F: H(m,n)

[rs]

=

• λ⊂[rs]

D˜

λ(r) · Dλ(s) · Z λ r|s · F (m,n) λ

(q, A)

Advances in Knot Polynomials 21 October 2016 35 / 49

. . .

❆ ❆ ✁ ✁ ✁ ✁ ❆ ❆

. . .

❍✟ ✟❍ ✟ ✟ ❍❍ ✯ ❥ ❑ ☛ ✕ ❯

2n 2m

Advances in Knot Polynomials 21 October 2016 36 / 49

Differential expansion for double braids

H(m,n)

[rs]

=

• λ∈[rs]

D˜

λ(r) · Dλ(s) · Z λ r|s · F (m,n) λ

(q, A) F (m,n)

λ

(q, A) ∼ F (m)

λ

(q, A) · F (n)

λ (q, A)

Advances in Knot Polynomials 21 October 2016 37 / 49

H(m,n)

[rs]

=

• λ⊂[rs]

D˜

λ(r) · Dλ(s) · Z λ r|s · F (m) λ

(q, A) · F (n)

λ (q, A)

F (1)

λ (q, A) · F (−1) λ

(q, A) F (m)

λ

(q, A) =

• µ∈λ

fλ,µ · Λ2m

µ

H(m,n)

R

=

• µ,ν∈R⊗¯

R

• DµDν

DR ¯ Sµν Λ2m

µ Λ2n ν

Advances in Knot Polynomials 21 October 2016 38 / 49

¯ S from DE for double braids

¯ Sµν = DR

• DµDν
• µ,ν⊂λ⊂R=[rs]

D˜

λ(r) · Dλ(s) · Z λ r|s

F (−1)

λ

(q, A) · fλ,µ · fλ,ν ¯ T ¯ S ¯ T = ST −1S† S =?

Advances in Knot Polynomials 21 October 2016 39 / 49

What are fλ,µ? ⇐ = DE for twist knots Tw(m)

m . . . −3 81 −2 61 −1 41 figure eight unknot 1 31 trefoil 2 52 3 72 . . . H(m)

[rs] =

• λ⊂[rs]

D˜

λ(r) · Dλ(s) · Z λ r|s · F (m) λ

(q, A) F (m)

λ

(q, A) =

• µ⊂λ

fλ,µ · Λ2m

µ

Advances in Knot Polynomials 21 October 2016 40 / 49

F (m)

λ

(q, A) =

µ⊂λ fλ,µ · Λ2m µ

F[1] = 1 − A2m 1 − A2 = A 1 {A} − A2m {A}

• F[2] = A
• 1

{Aq}{A} − [2] A2m {Aq2}{A} + (qA)4m {Aq}{Aq2}

• 1

[N]+ ? − → 1 [N] − 1 [N] = 0 1 [N + 1][N]+ ? − → 1 [N + 1][N] − [2] [N + 2][N] + 1 [N + 2][N + 1]

Advances in Knot Polynomials 21 October 2016 41 / 49

F (m)

λ

(q, A) =

µ⊂λ fλ,µ · Λ2m µ

• λ

1 [N + a′() − l′()]+ ? − →

• µ⊂λ

. . .

• λ∪µ[N + a′() − l′()]

Advances in Knot Polynomials 21 October 2016 42 / 49

F (m)

λ

(q, A) =

µ⊂λ fλ,µ · Λ2m µ

• λ

1 [N + a′() − l′()] + ? − →

• µ⊂λ

. . .

• λ∪µ[N + a′() − l′()]

Numerator nor factorized, e.g. f[3,2],[1] ∼ A2q8 + 2A2q6 + A2q4 + A2q2 − q6 − q4 − 2q2 − 1

Advances in Knot Polynomials 21 October 2016 43 / 49

F (m)

λ

(q, A) =

µ⊂λ fλ,µ · Λ2m µ

= ⇒ Skew Schur functions

f[3,2],[1] ∼ A2q8 + 2A2q6 + A2q4 + A2q2 − q6 − q4 − 2q2 − 1 ∼ χ∗

[3,2]/[1]

χλ{p′

k + p′′ k} =

• µ⊂λ

χλ/µ{p′

k} · χµ{p′′ k}

p∗

k = {Ak}

{qk} = [Nk] [k]

Advances in Knot Polynomials 21 October 2016 44 / 49

F (m)

λ

(q, A) =

µ⊂λ fλ,µ · Λ2m µ

= ⇒ Shifted skew Schur fns

fλ,µ ∼ χ∗

λ/µ · χ∗ µ

χ∗

λ

• A−

→A·q|µ|−1

Proportionality coefficient is a fully factorized product of differentials Explicitly known for R = [rr] or R = [2s]

Advances in Knot Polynomials 21 October 2016 45 / 49

Properties of HOMFLY

Polynomiality and integralities Factorizations Equations Hurwitz integrability Vogel’s universality (unification of E8-sectors of all Lie algebras) Differential expansions . . .

Advances in Knot Polynomials 21 October 2016 46 / 49

MANY THANKS FOR YOUR ATTENTION!

Advances in Knot Polynomials 21 October 2016 47 / 49

MANY THANKS to the ORGANIZERS!

Advances in Knot Polynomials 21 October 2016 48 / 49

VIVE l’AMITI´ E Franco-Russe!

Advances in Knot Polynomials 21 October 2016 49 / 49