Paper · Preprint · version 5 of 2026-11-04, first deposited 2026-10-05

The degree of the Alexander polynomial of twisted torus knots for every s

Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)

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Permanent archive (all files, all versions): doi:10.5281/zenodo.23155383 (Zenodo, CERN).

Abstract

For the twisted torus knot K(p,q;r,s), the closure of delta_p^q delta_r^(rs) with 0 < q < p, gcd(p,q) = 1, 2 <= r < p and s != 0, we give a closed formula for the degree (highest minus lowest exponent) of the Alexander polynomial, valid for every s. It is expressed through the degree sequence (d_m) of our earlier paper on the leading coefficient: deg Delta = 2 max_m (d_m + m r s) - (q+1)(p-1) - (r-1) r s. For s < 0, Park and Adnan determined it in two regimes. The highest exponent comes from the earlier paper, whose two theorems used here are re-proved in an appendix; the lowest follows from the symmetry of Delta, whose exact centre is fixed by the exponent sum of the braid. This note is a companion to that paper and closes the question its Remark 9 leaves open. The deposit contains the note (PDF and LaTeX source) and the scripts, data files and outputs behind its Section "Checks". Version 5 (4 November 2026) adds a remark on the degree formula of Bryant (arXiv:2611.03265v1, by Dehn surgery, for 1 < q < p): it agrees with ours (both are proved; a regrouping of his Theorem 1.3 into the multiset law of the companion paper shows it directly), and as integer formulas on 371,200 cases (one new script); nothing from it is used.

Keywords: knot theory · Alexander polynomial · twisted torus knots · braids · Burau representation · degree

Cite

@misc{stanggassinger2026alexander,
  author    = {Stanggassinger, Werner Alois},
  title     = {The degree of the Alexander polynomial of twisted torus knots for every s},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23155383},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/alexander-degree-twisted-torus-knots/}
}

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