Algebra of quantum $\mathcal{C}$-polynomials

Abstract

Knot polynomials colored with symmetric representations of $S{L}_q(N)$ satisfy difference equations as functions of representation parameter, which look like quantization of classical $\mathcal{A}$-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum $\mathcal{C}$-polynomials. It turns out that, for each knot, one can actually derive two difference equations of a finite order for these coefficients, those with shifts in spin $n$ of the representation and in $A=q^N$. Thus, the $\mathcal{C}$-polynomials are much richer and form an entire ring. We demonstrate this with the examples of various defect zero knots, mostly discussing the entire twist family.

arXiv comment21 pages

References in corpus (42)

• Khovanov's homology for tangles and cobordisms
• Knot Invariants and Topological Strings
• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Integrability and Matrix Models
• On Khovanov's categorification of the Jones polynomial
• Matrix factorizations and link homology II
• Knots, links and branes at large N
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Polynomial invariants for torus knots and topological strings
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• The colored Jones function is q-holonomic
• sl(3) link homology
• $2d$ gravity and matrix models. I. $2d$ gravity
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Super-A-polynomial for knots and BPS states
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Evolution method and "differential hierarchy" of colored knot polynomials
• Volume Conjecture: Refined and Categorified
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• Towards effective topological field theory for knots
• Differential hierarchy and additional grading of knot polynomials
• Topological Strings, D-Model, and Knot Contact Homology
• Tabulating knot polynomials for arborescent knots
• Genus expansion of HOMFLY polynomials
• Equations on knot polynomials and 3d/5d duality
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• Computing Khovanov-Rozansky homology and defect fusion
• On genus expansion of superpolynomials
• On the defect and stability of differential expansion
• Differential expansion and rectangular HOMFLY for the figure eight knot
• The colored HOMFLYPT function is $q$-holonomic
• Nimble evolution for pretzel Khovanov polynomials
• Tangle blocks in the theory of link invariants
• The C-polynomial of a knot
• Perspectives of differential expansion
• Knot polynomials for twist satellites
• Challenges of Matrix Models
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Large Nc gauge theory with quarks in high representations

Cited by in corpus (4)

• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Defect and degree of the Alexander polynomial
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket