2021/08/27 by Efstratia Kalfagianni, Kalfagianni, Efstratia, Christine Ruey Shan Lee +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #57K10 #57K14 #57K16 #Botulinum Toxin and Related Neurological Disorders #Connective tissue disorders research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2108.12391
openalex publication_date 2021/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate. As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image. We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot.