The colored HOMFLY polynomial is q-holonomic

Stavros Garoufalidis

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The colored HOMFLY polynomial is q-holonomic

Stavros Garoufalidis

Geometric Topology · Submitted 27 Nov 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.09 · mentions code or data availability 0.02.

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We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or the topological recursion, and motivates questions on the web approach to representation theory.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters.	result 0.81	0.96	0.96	asserted 1.00
s1	As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space.	result 0.95	0.93	0.12	asserted 1.00
s2	Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or the topological recursion, and motivates questions on the web approach to representation theory.	interpretation 0.99	0.40	0.04	asserted 0.66

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The paper

Original abstract

We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or the topological recursion, and motivates questions on the web approach to representation theory.

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References

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Topological Strings, D-Model, and Knot Contact Homology
hep-th 2013-04 conditional novelty 8.0 of 10

The Gromov-Witten disk potential of a knot's conormal Lagrangian and the augmentation variety of its contact homology DGA are shown, modulo stated conjectures, to define the same algebraic curve, with a D-module refor...
K-Decompositions and 3d Gauge Theories
hep-th 2013-01 conditional novelty 7.0 of 10

A hypersimplicial K-decomposition of ideal triangulations yields octahedron gluing equations that construct framed flat PGL(K,C) moduli spaces and candidate 3d N=2 theories T_K[M].

How the literature uses it

14 arXiv papers cite this in 18 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 15 sentences without a confident label

Uses its method or tool 1 paper

Then the recursion relation can be computed algorithmically [ref] , see [ref] .

q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links · Introduction and statement of results · 0.89

Cites it as support 1 paper

A strong piece of evidence for our conjecture is the fact, recently proven in [ref] , that the HOMFLY polynomial of a link $H(K)_m_1, …, m_$ colored by totally symmetric representations generates a D-module for the Weyl algebra $D$, acting by [equation] and [equation] whose characteristic variety is a Lagrangian submanifold.

Topological Strings, D-Model, and Knot Contact Homology · $D$-modules and quantization: the D-model and the D-mirror · 1.00

Cites it as background 1 paper

The colored HOMFLY polynomials are observed to satisfy linear recurrence relations with respect to representation [ref] known as quantum A-polynomials.

Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed · New $C$-polynomial for the twist knots · 0.90

What it cites · 16 sentences without a confident label

Uses the method or tool of 2 papers

Our description of the Jeong-Kim algorithm is implicit in the recent work of [ref] and uses only the relations (2.3)-(2.8) of [ref] .

Webs and quantum skew Howe duality · MOY graphs · 0.89
An algorithm for evaluating MOY graphs is described in [ref] .

The quantum $\mathfrak{sl}(n,\mathbb{C})$ representation theory and its applications · MOY graphs · 0.83

Compares with 1 paper

rem.cases Theorem [ref] was previously known for torus links in [ref] , and for finitely many twist knots (that include the $3_1, 4_1, 5_2$ and $6_1$ knots) in [ref] .

Super-A-polynomials for Twist Knots · Our results · 0.84

Cites as background 8 papers

Another super-polynomial (an element of $[q^±1,a^±1,t^±1][M] L$, where $LM=qML$) that specializes to a recursion for the sequence $(J^_N_K, nł(q))_n ∈$ when $t=-1$ and $a=q^N$ was conjectured to exist in [ref] .

Homological algebra of knots and BPS states · Super-polynomials in mathematical physics · 0.84
Another super-polynomial (an element of $[q^±1,a^±1,t^±1][M] L$, where $LM=qML$) that specializes to a recursion for the sequence $(J^_N_K, nł(q))_n ∈$ when $t=-1$ and $a=q^N$ was conjectured to exist in [ref] .

Volume Conjecture: Refined and Categorified · Super-polynomials in mathematical physics · 0.84
Another super-polynomial (an element of $[q^±1,a^±1,t^±1][M] L$, where $LM=qML$) that specializes to a recursion for the sequence $(J^_N_K, nł(q))_n ∈$ when $t=-1$ and $a=q^N$ was conjectured to exist in [ref] .

Super-A-polynomial for knots and BPS states · Super-polynomials in mathematical physics · 0.84

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