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A Survey of Quantum Enhancements

Sam Nelson

Abstract

In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.

A Survey of Quantum Enhancements

Abstract

In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.

Paper Structure

This paper contains 5 sections, 3 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Biquandles and Other Coloring Structures
  3. Biquandle Brackets
  4. Other Quantum Enhancements
  5. Questions

Key Result

Theorem 1

Given an oriented link diagram $D$ with a coloring by a biquandle $X$, for any diagram $D'$ obtained from $D$ by a single Reidemeister move, there is a unique coloring of $D'$ by $X$ which agrees with the coloring on $D$ outside the neighborhood of the move.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Example 1
  • Example 2
  • Example 3
  • Definition 2
  • Definition 3
  • Theorem 3
  • Example 4
  • ...and 2 more