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Biquandle Fares and Link Invariants

Sam Nelson, Stella Shah

TL;DR

This paper introduces biquandle fares, a new infinite family of invariants for oriented classical and virtual knots and links, built from path-like routes in biquandle-colored diagrams mapped into an abelian group. By defining fares of order $n$ and enforcing Reidemeister-move invariance, the authors obtain enhanced invariants in multiset or polynomial form, with detailed treatment of $n=1$ and $n=2$ cases, including complete, through, and crooked $2$-fares. They provide explicit examples demonstrating that these invariants can distinguish knots with identical biquandle counting invariants and show the computational viability via concrete computations. They also discuss decomposability of $2$-fares and raise questions about higher-order fares, potential homology-theoretic structures, and relationships to existing invariants such as biquandle brackets.

Abstract

We introduce a new family of invariants of oriented classical and virtual knots and links using fares, maps from paths in biquandle-colored diagrams to an abelian coefficient group. We consider the cases of 1-fares and 2-fares, provide examples to show that the enhancements are proper and end with some open questions about the cases of n-fares for n > 2.

Biquandle Fares and Link Invariants

TL;DR

This paper introduces biquandle fares, a new infinite family of invariants for oriented classical and virtual knots and links, built from path-like routes in biquandle-colored diagrams mapped into an abelian group. By defining fares of order and enforcing Reidemeister-move invariance, the authors obtain enhanced invariants in multiset or polynomial form, with detailed treatment of and cases, including complete, through, and crooked -fares. They provide explicit examples demonstrating that these invariants can distinguish knots with identical biquandle counting invariants and show the computational viability via concrete computations. They also discuss decomposability of -fares and raise questions about higher-order fares, potential homology-theoretic structures, and relationships to existing invariants such as biquandle brackets.

Abstract

We introduce a new family of invariants of oriented classical and virtual knots and links using fares, maps from paths in biquandle-colored diagrams to an abelian coefficient group. We consider the cases of 1-fares and 2-fares, provide examples to show that the enhancements are proper and end with some open questions about the cases of n-fares for n > 2.
Paper Structure (4 sections, 2 theorems, 30 equations)

This paper contains 4 sections, 2 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Biquandles and Homsets
  3. Biquandle Fares
  4. Questions

Key Result

Theorem 1

Let $X$ be a biquandle, $L$ an oriented classical or virtual knot, link multi-knotoid, $A$ an abelian group and $\phi:X\to A$ a biquandle $1$-fare. Then the multiset of fare values over the biquandle homset is invariant under Reidemeister moves.

Theorems & Definitions (21)

  • Definition 1
  • Example 1
  • Example 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Example 3
  • Definition 5
  • ...and 11 more