The crossing number of composite knots

TL;DR

This paper investigates the additivity of crossing numbers in knot theory, proving bounds that relate the crossing number of a connected sum of knots to the sum of their individual crossing numbers.

Contribution

It establishes new bounds showing the crossing number of a connected sum is at most the sum and at least a fraction of it, advancing understanding of knot complexity.

Findings

Proves c(K#K') ≤ c(K) + c(K')

Establishes c(K#K') ≥ (c(K) + c(K'))/152

Provides bounds supporting the conjecture of additivity

Abstract

It is a very old conjecture that the crossing number of knots is additive under connected sum. In other words, if K#K' is the connected sum of knots K and K', then does the equality c(K#K') = c(K) + c(K') hold? We prove that c(K#K') is at most c(K) + c(K') and at least (c(K) + c(K'))/152.