A Geometric Knotspace Template

TL;DR

This paper introduces a compact, geometrically-inspired polygonal tessellation template for knotspaces derived directly from knot diagrams, enabling efficient analysis of 3D knots and links.

Contribution

It provides the first explicit, geometrically-based tessellation template for knotspaces directly from knot diagrams, improving upon previous complex methods.

Findings

Template constructed from knot diagram with O(C) complexity

Allows derivation of a novel fundamental group presentation

Enables new measures of knot diagram complexity

Abstract

Early last century witnessed both the complete classification of 2-dimensional manifolds and a proof that classification of 4-dimensional manifolds is undecidable, setting up 3-dimensional manifolds as a central battleground of topology to this day. A rather important subset of the 3-manifolds has turned out to be the knotspaces, the manifolds left when a thin tube around a knot in 3D space is excised. Given a knot diagram it would be desirable to provide as compact a description of its knotspace as feasible; hitherto this has been done by computationally tessellating the knotspace of a given knot into polyhedral complexes using ad hoc methods of uncontrolled computational complexity. Here we present an extremely compact representation of the knotspace obtainable directly from a knot diagram; more technically, an explicit, geometrically-inspired polygonal tessellation of a deformation retract of the knotspace of arbitrary knots and links. Our template can be constructed directly from a planar presentation of the knot with C crossings using at most 12C polygons bounded by 64C edges, in time O(C). We show the utility of our template by deriving a novel presentation of the fundamental group, from which we motivate a measure of complexity of the knot diagram.