Towards effective topological field theory for knots

Abstract

Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds additional light on the structure of those mysterious formulas. Explicit expressions are now combined from Racah matrices of the type $R\otimes R\otimes \bar{R}⟶\bar{R}$ and mixing matrices in the sectors $R^{\otimes 3}⟶Q$. Further extension is provided by composition rules, allowing to glue two blocks, connected by an m-strand braid (they generalize the product formula for ordinary composite knots with m=1).

10 pages + table in Appendix

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Cited by in corpus (38)

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• Tabulating knot polynomials for arborescent knots
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Tangle blocks in the theory of link invariants
• Differential expansion for link polynomials
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• On R-matrix approaches to knot invariants
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Eigenvalue conjecture and colored Alexander polynomials
• Rectangular superpolynomials for the figure-eight knot
• Are there p-adic knot invariants?
• Distinguishing Mutant Knots
• Perspectives of differential expansion
• Checks of integrality properties in topological strings
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Gaussian distribution of LMOV numbers
• On matrix-model approach to simplified Khovanov-Rozansky calculus
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Algebra of quantum $\mathcal{C}$-polynomials
• SU(2)/SL(2) knot invariants and KS monodromies
• Defect and degree of the Alexander polynomial
• On geometric bases for quantum A-polynomials of knots
• Invariants of knots and links at roots of unity
• Vogel's universality and the classification problem for Jacobi identities
• Matrix model and dimensions at hypercube vertices
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Colored HOMFLY and Generalized Mandelbrot set
• On ambiguity in knot polynomials for virtual knots