The half-twisted splice operation on reduced knot projections

TL;DR

This paper demonstrates that all nontrivial reduced knot projections can be generated from a trefoil projection through a finite sequence of half-twisted splice operations, maintaining reduced projections at each step.

Contribution

It introduces the half-twisted splice operation as a method to generate all nontrivial reduced knot projections from a trefoil.

Findings

Any nontrivial reduced knot projection can be obtained from a trefoil by half-twisted splices.

Each step in the sequence preserves the reduced property.

The process is finite and constructive.

Abstract

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.