The Fox Trapezoidal Conjecture for Alternating Knots

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Abstract

A long-standing conjecture due to R. Fox states that the coefficients of the Alexander polynomial of an alternating knot exhibit a trapezoidal pattern. In other words, these coefficients increase, stabilize, then decrease in a symmetric way. A stronger version of this conjecture states that these coefficients form a log-concave sequence. This conjecture has been recently highlighted by J. Huh as one of the most interesting problems on log-concavity of sequences. In this expository paper, we shall review the various versions of the conjecture, highlight settled cases and outline some future directions.

Original languageEnglish
Article number495
JournalSymmetry
Volume16
Issue number4
DOIs
Publication statusPublished - Apr 2024

Keywords

  • Alexander polynomial
  • alternating knot
  • log-concave
  • trapezoidal conjecture

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)

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