Unknotting number is not additive under connected sum

TL;DR

This paper provides the first examples showing that the unknotting number of a connected sum of knots can be strictly less than the sum of their individual unknotting numbers, answering a longstanding open question.

Contribution

It introduces the first known examples where unknotting number is not additive under connected sum, resolving a problem from Kirby's list.

Findings

Unknotting number is not additive under connected sum.

Provides explicit examples of knots with non-additive unknotting numbers.

Answers a question from Kirby's problem list in low-dimensional topology.

Abstract

We give the first examples of a pair of knots $K_1$,$K_2$ in the 3-sphere for which their unknotting numbers satisfy $u(K_1\#K_2)<u(K_1)+u(K_2)$ . This answers question 1.69(B) from Kirby's problem list, "Problems in low-dimensional topology", in the negative.