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Generalized knots-quivers correspondence

Marko Stošić

Abstract

We propose a generalized version of knots-quivers correspondence, where the quiver series variables specialize to arbitrary powers of the knot HOMFLY-PT polynomial series variable. We explicitely compute quivers for large classes of knots, as well as many homologically thick 9- and 10-crossings knots, including the ones with the super-exponential growth property of colored HOMFLY-PT polyomials. In addition, we propose a new, compact, quiver-like form for the colored HOMFLY-PT polynomials, where the structure of colored differentials is manifest. In particular, this form partially explains the non-uniqueness of quivers corresponding to a given knot via knots-quivers correspondence.

Generalized knots-quivers correspondence

Abstract

We propose a generalized version of knots-quivers correspondence, where the quiver series variables specialize to arbitrary powers of the knot HOMFLY-PT polynomial series variable. We explicitely compute quivers for large classes of knots, as well as many homologically thick 9- and 10-crossings knots, including the ones with the super-exponential growth property of colored HOMFLY-PT polyomials. In addition, we propose a new, compact, quiver-like form for the colored HOMFLY-PT polynomials, where the structure of colored differentials is manifest. In particular, this form partially explains the non-uniqueness of quivers corresponding to a given knot via knots-quivers correspondence.
Paper Structure (37 sections, 5 theorems, 149 equations, 1 figure)

This paper contains 37 sections, 5 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. LMOV conjecture
  4. Special form of colored HOMFLY-PT polynomials
  5. Rewriting ansatz as true quiver generating series
  6. Non-uniqueness
  7. Explicit results
  8. $\mathbf{9_{42}}$
  9. ${\bf 9_{46}}$
  10. $\bf 8_{20}$
  11. $\mathbf{10_{132}}$
  12. ${\bf 10_{145}}$
  13. ${\bf 10_{139}}$
  14. ${\bf 10_{152}}$
  15. $(2,2p+1)$ torus knots
  16. ...and 22 more sections

Key Result

Theorem 2

Ef$\Omega_{d_1,\ldots,d_m;j}$ are nonnegative integers.

Figures (1)

  • Figure 1: Quiver matrix for $\mathbf{9_{42}}$

Theorems & Definitions (10)

  • Conjecture 1: Generalized knots-quivers correspondence
  • Definition 1: Quiver-form of colored HOMFLY-PT polynomials
  • Definition 2: Generalized quiver-form of colored HOMFLY-PT polynomials
  • Theorem 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Proposition 2
  • Proposition 3
  • Proposition 4