2008/05/31 by Marc Lackenby · 20 citations
Mathematics · #Advanced Combinatorial Mathematics #Composite number #Conjecture #Crossing number (knot theory) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.1112/jtopol/jtp028
published in Journal of Topology 2(4), 747-768 (Wiley) · 28 pages, 20 figures; final version, to appear in the Journal of Topology
openalex publication_date 2009/01/01 · arxiv created 2009/08/25 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
It is a very old conjecture that the crossing number of knots is additive under connected sum. In other words, if K1 ♯ K2 is the connected sum of knots K1 and K2, then does the equality c(K1 ♯ K2) = c(K1) + c(K2) hold? We prove that c(K1 ♯ K2) is at most c(K1) + c(K2) and at least (c(K1) + c(K2))/152.