The Alexander polynomial of twisted torus knots in the complementary range

@misc{adnan2026alexander,
  title = {{The Alexander polynomial of twisted torus knots in the complementary range}},
  author = {Adnan and Kyungbae Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34813},
  eprint = {2610.34813},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34813},
}

TY  - UNPB
TI  - The Alexander polynomial of twisted torus knots in the complementary range
AU  - Adnan
AU  - Kyungbae Park
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34813
UR  - https://arxiv.org/abs/2610.34813
AB  - A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}<r<=|p|+|q| by placing the knot on a standard genus-two Heegaard surface and applying Fox calculus. We then combine the two calculations into a unified Alexander-polynomial formula valid for 1<r<=|p|+|q|. In the complementary range, we prove monicity and give an explicit formula for the breadth. This determines the Seifert genus for nonnegative twists and gives a genus lower bound for negative twists. We also exhibit infinite two-parameter families of non-L-space knots in the complementary range.
ER  -