Mirror-Curves and Knot Mosaics

TL;DR

This paper introduces mirror-curves as a concise and computationally efficient representation of knots and links, connecting knot mosaics, grid diagrams, and quantum knot theory, with algorithms for polynomial invariants.

Contribution

It develops a new mirror-curve coding system for knots and links, providing minimal codes and algorithms for computing knot invariants directly from these representations.

Findings

Tables of minimal mirror-curve codes for small knots and links

An efficient algorithm for computing Kauffman bracket and L-polynomials from mirror-curves

Establishment of mirror-curves as a unifying framework for knot representations

Abstract

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams [3,7,22,24]. Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links obtained from rectangular grids of size 3x3 and px2 (p<5), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials [18,19,20] directly from mirror-curve representations.