Additivity of Crossing Number via Restricted Reidemeister Moves

TL;DR

This paper introduces restricted Reidemeister moves and proves that crossing numbers are additive under these moves for knot sums, with additional topological insights.

Contribution

It establishes a new set of restricted Reidemeister moves and proves crossing number additivity for knot sums using these moves.

Findings

Crossing number of a knot obtained from a sum is at least the sum of individual crossing numbers.

Restricted Reidemeister moves preserve certain crossing number properties.

Topological interpretations provide deeper understanding of crossing number additivity.

Abstract

We define a set of restricted Reidemeister moves and show that if $K$ is obtained from $K_0\,\#\,K_1$ using those moves, then the crossing number of $K$ is at least $c(K_0)+c(K_1)$. We also explore topological interpretations of this result.