Grope metrics on the knot concordance set

TL;DR

This paper introduces a family of pseudo-metrics on the knot concordance set based on admissible grope lengths, revealing non-discrete topologies and connecting to knot invariants, supporting the fractal conjecture of the concordance group.

Contribution

It defines new grope-based metrics on knot concordance classes and analyzes their properties, including non-discreteness and relations to existing invariants, advancing understanding of the group's structure.

Findings

The topology from these metrics is non-discrete for all q>1.

Winding number zero satellite operators are contractions under these metrics.

Knot signatures provide lower bounds for the metrics.

Abstract

To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q>1. The analogous statement for links also holds for q=1. In addition we translate much previous work on knot concordance into distance statements. In particular, we show that winding number zero satellite operators are contractions in many cases, and we give lower bounds on our metrics arising from knot signatures and higher order order signatures. This gives further evidence in favor of the conjecture that the knot concordance group has a fractal structure.