Cobordism distance on the projective space of the knot concordance group

TL;DR

This paper introduces a new metric called cobordism distance on the knot concordance group, leading to a projective space structure, and explores its properties using torus and twist knots.

Contribution

It defines the cobordism distance and the projective space of the knot concordance group, analyzing their properties and structure with specific classes of knots.

Findings

The projective space has an integer-valued metric.

The simplicial complex contains arbitrarily large simplices.

Basic properties of the metric are established.

Abstract

The cobordism distance on the knot concordance group is used to define a measure of how close two knots are to being linearly dependent. Roughly stated, d(K,J) is defined by minimizing the cobordism distance between pairs of knots in cyclic subgroups containing K and J. When made precise, this leads to the definition of the projective space of the knot concordance group. We explore basic properties of this projective space and its integer-valued metric by considering torus knots. Twist knots are used to show that the associated simplicial complex contains an infinite set of simplices of arbitrarily large dimension.