Minimal Generating Sets of Singular Reidemeister Moves and Their Classification

TL;DR

This paper completely classifies minimal generating sets of singular Reidemeister moves in singular knot theory, introducing a new invariant to establish independence and enumeration results.

Contribution

It provides a full classification of minimal generating sets for singular Reidemeister moves and introduces a new invariant to prove move independence.

Findings

Exactly 96 minimal generating sets for oriented singular Reidemeister moves.

The classification reduces to 8 minimal generating sets in the unoriented case.

A new invariant distinguishes move types and obstructs certain move combinations.

Abstract

Singular knot theory extends classical knot theory by allowing transverse double points without over/under information, together with singular Reidemeister moves of types IV and V. A central open problem in this theory is to determine the minimal generating sets of oriented singular Reidemeister moves. In this paper, we completely solve this problem. In addition, we establish independence results for singular Reidemeister moves by introducing an invariant that provides obstructions and lower bounds for generating sets, including the independence of type III from types I, II, IV, and V. More precisely, starting from a minimal generating set of ordinary Reidemeister moves of types I--III, we prove that the singular moves admit exactly $96$ distinct inclusion-minimal generating sets, and that these exhaust all possibilities. Our proof introduces a new invariant for singular links, constructed via a projection to self-singular links, which detects the distinction between the two families of type IV moves and provides an obstruction for generating type V moves from types I--IV. We also determine the unoriented case, where the classification collapses to exactly $8$ minimal generating sets.