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PreprintCC BY 4.0arXiv

Multivariable Alexander polynomials of twisted torus links via Dehn surgery

  • Birch Bryant
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Abstract

Let $T(p,q;r,s)$ be the twisted torus link obtained from the torus link T(p,q) by adding s full twists to r adjacent strands, and let C be an unknotted circle encircling those strands. For all p, q, and r we compute the multivariable Alexander polynomial of T(p,q)∪ C in closed form. A relation of Silver and Williams between the Alexander polynomials of a link and of its 1/n Dehn fillings then gives a closed formula for the multivariable Alexander polynomial of every twisted torus link $T(p,q;r,s)$, answering a question of Adnan and Park. When gcd(p,q)=1, so that $T(p,q;r,s)$ is a knot, the formula is a sum of pr monomials with coefficient +1, divided by cyclotomic factors, and its exponents are governed by an elementary counting function on the orbit of q modulo p. Because the numerator has no cancellation, we are able to give the degree for all s and a simple non-fiberedness criterion.

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Cite

BibTeXRIS
BibTeX
@misc{bryant2026multivariable,
  title = {{Multivariable Alexander polynomials of twisted torus links via Dehn surgery}},
  author = {Birch Bryant},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03265},
  eprint = {2611.03265},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03265},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Multivariable Alexander polynomials of twisted torus links via Dehn surgery
AU  - Birch Bryant
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03265
UR  - https://arxiv.org/abs/2611.03265
AB  - Let $T(p,q;r,s)$ be the twisted torus link obtained from the torus link T(p,q) by adding s full twists to r adjacent strands, and let C be an unknotted circle encircling those strands. For all p, q, and r we compute the multivariable Alexander polynomial of T(p,q)∪ C in closed form. A relation of Silver and Williams between the Alexander polynomials of a link and of its 1/n Dehn fillings then gives a closed formula for the multivariable Alexander polynomial of every twisted torus link $T(p,q;r,s)$, answering a question of Adnan and Park. When gcd(p,q)=1, so that $T(p,q;r,s)$ is a knot, the formula is a sum of pr monomials with coefficient +1, divided by cyclotomic factors, and its exponents are governed by an elementary counting function on the orbit of q modulo p. Because the numerator has no cancellation, we are able to give the degree for all s and a simple non-fiberedness criterion.
ER  -