Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

q-Rationals, link invariants and webs

Perrine Jouteur, Hoel Queffelec

Abstract

We investigate the role of q-rationals in the context of link invariants. This leads us to introduce and study web categories that q-deform Deligne's categories.

q-Rationals, link invariants and webs

Abstract

We investigate the role of q-rationals in the context of link invariants. This leads us to introduce and study web categories that q-deform Deligne's categories.
Paper Structure (15 sections, 15 theorems, 68 equations, 2 figures)

This paper contains 15 sections, 15 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $q$-real numbers
  3. Links and web categories
  4. Knot and links
  5. Upward webs
  6. A universal web category, and the HOMFLY-PT invariant
  7. Specializations and Reshetikhin-Turaev polynomials
  8. Interpolating web categories
  9. Definition and presentation
  10. Without the quadratic extension
  11. Webs, quantized Deligne's category and interpolation between categories of representations
  12. Structure of the specialization
  13. The web categories and related categories
  14. Behavior of the $x$-polynomial
  15. Left $q$-rationals

Key Result

Lemma 3.6

Colored twists act as follows:

Figures (2)

  • Figure 1: Computing the HOMFLY-PT polynomial
  • Figure 2: Value of the $x$-invariant for $x\in [0,1]$ and $q=2$. Top left: stabilized unknot. Top right: trefoil. Bottom left: knot $5_1$. Bottom right: ratio between the polynomials for $5_1$ and the trefoil. The fact that there still are discontinuities shows that they are not only controlled by a global factor, which is an incentive to further study these invariants.

Theorems & Definitions (50)

  • Definition 2.1: MGO-2020
  • Definition 2.3
  • Definition 2.4: MGOr
  • Remark 2.5
  • Definition 3.1: framed Reidemeister moves
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Lemma 3.6
  • ...and 40 more