Second bounded cohomology of knot quandles

TL;DR

This paper investigates the second bounded cohomology of knot quandles, establishing conditions for infinite dimensionality and demonstrating its ability to detect the unknot, with applications to link and quandle structures.

Contribution

It provides new criteria for the infinite dimensionality of second bounded cohomology of quandles and applies these to knot and link quandles, advancing understanding in knot theory and algebraic topology.

Findings

Second bounded cohomology of non-split link quandles is infinite dimensional.

Second bounded cohomology detects the unknot.

Infinite dimensionality for free product quandles with amenable factors.

Abstract

In this paper, we explore the bounded cohomology of quandles and its applications to knot theory. We establish two key results that provide sufficient conditions for the infinite dimensionality of the second bounded cohomology of quandles. The first condition involves a subspace of homogeneous group quasimorphisms on the inner automorphism group of the quandle, whereas the second condition concerns the vanishing of the stable commutator length on a subgroup of this inner automorphism group. As topological applications, we show that the second bounded cohomology of the quandle of any non-split link whose link group is non-solvable as well as the quandle of any split link, is infinite dimensional. From these results, we conclude that the second bounded cohomology of the knot quandle detects the unknot. On the algebraic side, we prove that the second bounded cohomology of a free product of quandles is infinite dimensional if the inner automorphism group of at least one of the free factors is amenable. This leads to the result that the second bounded cohomology of free quandles of rank greater than one, as well as their canonical quotients, is infinite dimensional.