A lower bound of the crossing number of composite knots

TL;DR

This paper establishes a new lower bound for the crossing number of composite knots, showing it exceeds one-sixteenth of the sum of the crossing numbers of the component knots, advancing understanding of knot complexity.

Contribution

The paper provides the first nontrivial lower bound for the crossing number of composite knots, improving previous knowledge about their minimal crossing number.

Findings

Proves that c(K1#K2) > (c(K1)+c(K2))/16

Establishes a quantitative lower bound for composite knot crossing numbers

Advances the theoretical understanding of knot complexity relationships

Abstract

Let $c(K)$ denote the crossing number of a knot $K$ and let $K_1\# K_2$ denote the connected sum of two oriented knots $K_1$ and $K_2$. It is a very old unsolved question that whether $c(K_1\# K_2)=c(K_1)+c(K_2)$. In this paper we show that $c(K_1\# K_2)> (c(K_1)+c(K_2))/16$.