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: � � � H K R ( A , q ) = Tr R P exp A K average with CS action with the gauge group G = SL ( N ) : � � � A d A + 2 3 A 3 κ Tr M 2 π i q = exp κ + N A = q N 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 ( R 3 or S 3 ): 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
other 2d 3d higher d Seifert surfaces CFT ✟✟✟✟ ✯ ❅ ■ ✻ � ✒ ❅ � CS ❄ projection to 2d lattice theory on arbitrary graphs � ❅ ✠ � ❅ ❘ algebra geometry RT KhR hypercube Advances in Knot Polynomials 21 October 2016 9 / 49
RT calculus traditional ✻ ✻ tr q ρ . . . R ± 1 ✻ 34 R ± 1 modern 23 ✻ � Q D Q · Tr W Q . . . R ± 1 12 ✻ R k l † R Q 23 = U Q 23 R Q 12 U Q 23 i j � � R Q 12 = diag ǫ Y · q κ Y ✻ R R ⊗ R ⊗ R ⊗ R = ⊕ Q ⊗ W Q ✻ ✻ � �� � R R ⊕ Y Advances in Knot Polynomials 21 October 2016 10 / 49
Fundamental representation R = � : R − R − 1 = q − q − 1 skein relations 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 ) ⊗ R − → Q − → R ⊗ ( R ⊗ R ) − → Q � �� � � �� � Y ′ Y ′′ 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 � � � � ¯ ( R ⊗ ¯ R ⊗ (¯ S : 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 − 1 S † T and ¯ T are diagonal matrices Advances in Knot Polynomials 21 October 2016 18 / 49
FINGERED BRAIDS ✲ ✲ ✲ ✲ ✲ ✲ n 1 n 2 n 4 n 6 ✲ ✲ ✲ ✲ ✲ ✲ ✲ n 3 n 5 n 7 ✲ ✲ ✲ ✲ ✛ 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 σ AB − → σ X αβ with the gauge group acting by two orthogonal matrices A and B : � σ X ,α,β − → A αα ′ B ββ ′ σ X ,α ′ β ′ α ′ β ′ Advances in Knot Polynomials 21 October 2016 22 / 49
Quadratic terms in the Lagrangian are: "local"ones ϕ X ¯ σ X T n X σ X = σ X ,αβ T n T 2 n X ,αα ′ σ X ,α ′ β X ϕ X , φ X ¯ ϕ X ¯ φ X ¯ T 2 n T 2 n − 1 T 2 n X φ X , φ X , X ϕ X X plus conjugates, "non-local"ones X S † X ¯ σ ∗ φ ∗ ϕ ∗ XY φ Y , X S XY σ Y , S XY ϕ Y (note that there are no terms φ ∗ X φ Y ) Advances in Knot Polynomials 21 October 2016 23 / 49
Topologically allowed vertices are Γ (1) ∼ σ 3 Γ (2) ∼ ϕ 3 Γ (3) ∼ φ 2 X , X , X ϕ X The problem is, however, to deal with the Greek indices in Γ α,β,γ Φ α,β Φ β,γ Φ γ,α . A naive anzatz like tr σ 3 X with the trace in Greek → A σ A † , but it indices would be good for a transformation law σ − violates σ − → A σ B 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
2 n ❆ ✁ ☛ ❑ ❆ ✁ . . . ✁ ❆ ✁ ❆ ❯ ✕ ❍✟ 2 m . . . ✟ ✟❍ ❍❍ ❥ ✯ ✟ Advances in Knot Polynomials 21 October 2016 25 / 49
EVOLUTION for twist and double braid knots � D µ D ν � H ( m , n ) S µν Λ 2 m ¯ µ Λ 2 n = ν R D R µ,ν ∈ R ⊗ ¯ R Advances in Knot Polynomials 21 October 2016 26 / 49
Properties of HOMFLY Polynomiality and integralities Factorizations Equations Hurwitz integrability Vogel’s universality (unification of E 8 -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
[1] = A 2 − q 2 + 1 − q − 2 + A − 2 H 4 1 = 1 + { Aq }{ A / q } A = q N { x } = x − 1 / x [ n ] = q n − q − n q − q − 1 = { q n } { q } Differentials { Aq n } ∼ [ N + n ] Advances in Knot Polynomials 21 October 2016 30 / 49
H 4 1 [1] = 1 + { Aq }{ A / q } H 4 1 [2] = 1 + [2] { Aq 2 }{ A / q } + { Aq 3 }{ Aq 2 }{ A }{ A / q } H 4 1 [3] = 1 + [3] { Aq 3 }{ A / q } + [3] { Aq 4 }{ Aq 3 }{ A }{ A / q } + + { Aq 5 }{ Aq 4 }{ Aq 3 }{ Aq }{ A }{ A / q } . . . Advances in Knot Polynomials 21 October 2016 31 / 49
Differential expansion Equations . . . Superpolynomials տ ↑ ր H 4 1 [1] = 1 + { Aq }{ A / q } H 4 1 [2] = 1 + [2] { Aq 2 }{ A / q } + { Aq 3 }{ Aq 2 }{ A }{ A / q } ւ ց Other representations Other knots [ r s ] = � H 4 1 λ ( r ) D λ ( s ) Z λ H K [1] = 1 + G K λ ∈ [ r s ] D ˜ [1] ( q , A ) { Aq }{ A / q } r | s Advances in Knot Polynomials 21 October 2016 32 / 49
� { Aq r + a ′ ( � ) − l ′ ( � ) }{ Aq − s + a ′ ( � ) − l ′ ( � ) } Z λ r | s ( A , q ) = � ∈ λ l ′ ✻ ✛ ✲ a ′ a ❄ l { Aq − l ′ ( � )+ a ′ ( � ) } [ N − l ′ ( � ) + a ′ ( � )] � � D λ ( N ) = { q a ( � )+ l ( � )+1 } = [ a ( � ) + l ( � ) + 1] � ∈ λ � ∈ λ Advances in Knot Polynomials 21 October 2016 33 / 49
Other knots: H K [1] = 1 + G K [1] ( q , A ) { Aq }{ A / q } H K [2] = 1 + [2] G K [1] ( q , A ) { Aq 2 }{ A / q } + G K [2] ( q , A ) { Aq 2 }{ A }{ A / q } only for defect zero : [1] ( q , A ) { Aq 2 }{ A / q } + F K 0 H K [2] = 1 + [2] F K [2] ( q , A ) { Aq 3 }{ Aq 2 }{ A }{ A / q } � � Al K defect = power q 2 - 1 [1] Advances in Knot Polynomials 21 October 2016 34 / 49
Double braids have defect zero and very special F : � H ( m , n ) r | s · F ( m , n ) λ ( r ) · D λ ( s ) · Z λ = D ˜ ( q , A ) [ r s ] λ λ ⊂ [ r s ] Advances in Knot Polynomials 21 October 2016 35 / 49
2 n ❆ ✁ ☛ ❑ ❆ ✁ . . . ✁ ❆ ✁ ❆ ❯ ✕ ❍✟ 2 m . . . ✟ ✟❍ ❍❍ ❥ ✯ ✟ Advances in Knot Polynomials 21 October 2016 36 / 49
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