The degenerate distributive complex is degenerate

TL;DR

This paper proves that the degenerate part of distributive homology for multispindles, especially quandles, is determined by normalized homology, providing explicit formulas and clarifying its role in algebraic knot theory.

Contribution

It establishes that degenerate homology is fully determined by normalized homology for multispindles and quandles, with explicit formulas, resolving a long-standing mystery in algebraic knot theory.

Findings

Degenerate homology is determined by normalized homology for multispindles.

Explicit K"unneth-type formula for degenerate part of homology.

Clarifies the role of degenerate quandle homology in knot theory.

Abstract

We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle $Q$, the degenerate homology of $Q$ is completely determined by the quandle homology of $Q$. For this case (and generally for two term homology of a spindle) we provide an explicit K\"unneth-type formula for the degenerate part. This solves the mystery in algebraic knot theory of the meaning of the degenerate quandle homology, brought over 15 years ago when the homology theories were defined, and the degenerate part was observed to be non trivial.