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CWR sequence of invariants of alternating links and its properties

Michal Jablonowski

TL;DR

The paper develops the $CWR$ invariant for alternating links, a sequence of two-variable polynomials that strengthens the prior $WRP$ invariant by incorporating richer diagram structure through weighted Tait graphs. It proves flype invariance, provides matrix and skein-based computation methods, and demonstrates that $CWR$ can distinguish knots that classical invariants like $HOMFLYPT$ and $Kauffman$ 2v/3v cannot in many cases. Key contributions include a formal definition via $G_B^*$ and $G_W^*$, additive behavior under connected sums, mutation and chirality analyses (with notable exceptions such as $K14a506$), and explicit matrix formulas for $CWR_2$ and $CWR_3$ along with recursive relations. The authors also deliver computational tables for knots up to eight crossings and provide broader data up to sixteen crossings, illustrating $CWR$ as a practical and powerful discriminant tool in knot theory, especially for alternating knots.

Abstract

We present the $CWR$ invariant, a new invariant for alternating links, which builds upon and generalizes the $WRP$ invariant. The $CWR$ invariant is an array of two-variable polynomials that provides a stronger invariant compared to the $WRP$ invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman $3$-variable, and Kauffman $2$-variable polynomials on specific knot examples. Additionally, we derive general recursive "skein" relations, and also specific formulas for the initial components of the $CWR$ invariant using weighted adjacency matrices of modified Tait graphs.

CWR sequence of invariants of alternating links and its properties

TL;DR

The paper develops the invariant for alternating links, a sequence of two-variable polynomials that strengthens the prior invariant by incorporating richer diagram structure through weighted Tait graphs. It proves flype invariance, provides matrix and skein-based computation methods, and demonstrates that can distinguish knots that classical invariants like and 2v/3v cannot in many cases. Key contributions include a formal definition via and , additive behavior under connected sums, mutation and chirality analyses (with notable exceptions such as ), and explicit matrix formulas for and along with recursive relations. The authors also deliver computational tables for knots up to eight crossings and provide broader data up to sixteen crossings, illustrating as a practical and powerful discriminant tool in knot theory, especially for alternating knots.

Abstract

We present the invariant, a new invariant for alternating links, which builds upon and generalizes the invariant. The invariant is an array of two-variable polynomials that provides a stronger invariant compared to the invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman -variable, and Kauffman -variable polynomials on specific knot examples. Additionally, we derive general recursive "skein" relations, and also specific formulas for the initial components of the invariant using weighted adjacency matrices of modified Tait graphs.
Paper Structure (12 sections, 7 theorems, 8 equations, 9 figures)

This paper contains 12 sections, 7 theorems, 8 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. An example
  4. An invariance
  5. Properties
  6. Generalization of the other invariants
  7. Connected sum
  8. Non-mutant knots
  9. Chirality
  10. Matrix formulae for $CWR_2$ and $CWR_3$
  11. Recursive relations
  12. Values for knots in tables

Key Result

Theorem 2.1

For any pair $D_1$ and $D_2$ of reduced alternating diagrams of a given alternating link, we have

Figures (9)

  • Figure 1: A nugatory crossing (left) and a checkerboard coloring (right).
  • Figure 2: The convention for positive and negative crossings.
  • Figure 3: Knot $K7a1$ with their graphs $G_B^*$ (blue) and $G_W^*$ (red).
  • Figure 4: A flype-move on the tangle $R$ moves the crossing $c$ from one side of $R$ to the other, while rotating $R$ by $180^\circ$ around the horizontal axis.
  • Figure 5: The general situation on graphs in flype-moves. Case I and II.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 5 more