Minimal generating sets of directed oriented Reidemeister moves

TL;DR

This paper proves that a specific set of 8 directed oriented Reidemeister moves is minimal for generating all such moves, and extends the result to links with at least two components, discussing related invariants.

Contribution

It establishes the minimality of a particular set of directed Reidemeister moves and introduces the concept of L-generating sets for links, with implications for knot invariants.

Findings

The set of 8 directed Polyak moves is minimal for generating all directed oriented Reidemeister moves.

The same set is minimal for any link with at least two components.

Discussion of knot invariants related to L-generating sets and move types.

Abstract

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $8$ directed Polyak moves $\{ \Omega{1a}^\uparrow, \Omega{1a}^\downarrow, \Omega{1b}^\uparrow, \Omega{1b}^\downarrow, \Omega{2a}^\uparrow, \Omega{2a}^\downarrow, \Omega{3a}^\uparrow, \Omega{3a}^\downarrow \}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least $2$ components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\Omega3$.