The KnotMosaics Package for SageMath

TL;DR

KnotMosaics is a SageMath package that enables construction, visualization, and analysis of knot mosaic diagrams, integrating with existing knot software for advanced computations.

Contribution

It introduces a comprehensive SageMath package for knot mosaics, including validation, visualization, and connection to knot invariants and homology computations.

Findings

Supports validation and tracing of knot mosaics

Enables computation of Jones polynomials and knot Floer homology

Provides tools for generating random and rational tangle mosaics

Abstract

We introduce KnotMosaics, a SageMath package for constructing, visualizing, and analyzing knot mosaic diagrams. The package represents an n-mosaic as a matrix of standard tile labels and implements the local connectivity rules needed to validate mosaics, trace strands and components, compute planar diagram codes, generate random examples, and construct rational tangle mosaics. The planar diagram interface connects the mosaic representation to existing knot and link software, enabling computations such as Jones polynomials and knot Floer homology checks. We describe the package design, its main algorithms, and representative examples that illustrate how KnotMosaics can support computational exploration in knot mosaic theory.