On the homology of the space of singular knots

TL;DR

This paper explores the algebraic structures in the homology of spaces of knots and singular knots, establishing relations between products and demonstrating the nontriviality of desingularization maps.

Contribution

It introduces associative products on homology of knot spaces, relates them via desingularization, and computes key examples showing their nontriviality.

Findings

Established relations between associative products and desingularization maps.

Computed specific products in homology of knot spaces.

Proved the nontriviality of the desingularization morphism.

Abstract

In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.