The Kanenobu knots and Khovanov-Rozansky homology

TL;DR

This paper demonstrates that Kanenobu knots form an infinite family indistinguishable by both HOMFLY and sl(n) homologies, revealing limitations of these invariants in knot differentiation.

Contribution

It proves that Kanenobu knots share the same homologies, providing the first example of an infinite family undetectable by these invariants, based on a new structure theorem.

Findings

Kanenobu knots have identical HOMFLY and sl(n) homologies.

This invariance persists across an infinite family of knots.

The result highlights limitations of homological invariants in knot theory.

Abstract

Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.