Companions of the unknot and width additivity

2009/08/27 by Ryan Blair, Blair, Ryan, Maggy Tomova +1
Computer Science · Mathematics · #57M25 #57M27 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0908.4103

openalex publication_date 2009/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It has been conjectured that for knots K and K' in S³, w(K#K')= w(K)+w(K')-2. Scharlemann and Thompson have proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots Kⁿ_i for which they conjectured that w(Bⁿ#Kⁿ_i)=w(Kⁿ_i) where Bⁿ is a bridge number n knot. We show that for n>2 none of the knots in Kⁿ_i produces such counterexamples.

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