Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A polynomial pair invariant of alternating knots and links

Michal Jablonowski

TL;DR

This paper introduces $WRP$, an invariant for alternating knots and links formed as an unordered pair of integer polynomials in $w$ and $r$ derived from the two checkerboard graphs of a reduced diagram. It proves $WRP$ is well-defined and invariant under flypes, and provides explicit calculations including a closed form for $K_{3, a1}$ and general expressions for torus and twist knots; the invariant is then tabulated for prime knots up to $10$ crossings (extending to $13$ in supplementary material) and benchmarked against classical invariants. The results show $WRP$ distinguishes all prime alternating knots up to $10$ crossings (including mirror images) and often provides stronger discrimination than several established invariants, though it cannot resolve all mutants. Overall, $WRP$ offers a robust, diagrammatic, polynomial-based tool that complements existing knot invariants for classification and comparison in the alternating knot family.

Abstract

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket.

A polynomial pair invariant of alternating knots and links

TL;DR

This paper introduces , an invariant for alternating knots and links formed as an unordered pair of integer polynomials in and derived from the two checkerboard graphs of a reduced diagram. It proves is well-defined and invariant under flypes, and provides explicit calculations including a closed form for and general expressions for torus and twist knots; the invariant is then tabulated for prime knots up to crossings (extending to in supplementary material) and benchmarked against classical invariants. The results show distinguishes all prime alternating knots up to crossings (including mirror images) and often provides stronger discrimination than several established invariants, though it cannot resolve all mutants. Overall, offers a robust, diagrammatic, polynomial-based tool that complements existing knot invariants for classification and comparison in the alternating knot family.

Abstract

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket.
Paper Structure (8 sections, 1 theorem, 6 equations, 5 figures, 1 table)

This paper contains 8 sections, 1 theorem, 6 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Graphs
  4. Invariant
  5. Calculations
  6. Invariance
  7. Tabulation
  8. Comparisons to other knot invariants

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 (5)

  • Figure 1: Calculating $WRP$ of a trefoil.
  • Figure 2: A flype on the tangle $R$ moves the crossing $c$ from one side of $R$ to the other, while turning $R$ by $180^\circ$ around the horizontal axis.
  • Figure 3: Case I.
  • Figure 4: Case II.
  • Figure 5: Knots $K11a75$ and $K11a102$ with their graphs $G_B'$ and $G_W'$.

Theorems & Definitions (3)

  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['twA']}