PreprintarXiv licencearXiv
Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings
Abstract
Let K be a hyperbolic knot and let ρn be the n-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with ρn detect the complex volume of the knot complement. We compute π/2 log ( (AK,n−2(1)AK,n+2(1))/AK,n(1)2 ) for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach Vol(S3∖ K) +i 2π2CS(S3∖ K) modulo iπ2ℤ. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.
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BibTeX
@misc{aso2026asymptotic,
title = {{Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings}},
author = {Airi Aso},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29124},
eprint = {2609.29124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29124},
}
RIS
TY - UNPB
TI - Asymptotic behavior of twisted Alexander invariants for hyperbolic knots with at most six crossings
AU - Airi Aso
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29124
UR - https://arxiv.org/abs/2609.29124
AB - Let K be a hyperbolic knot and let ρ_n be the n-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with ρ_n detect the complex volume of the knot complement. We compute π/2 log ( (A_K,n−2(1)A_K,n+2(1))/A_K,n(1)^2 ) for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach Vol(S^3∖ K) +i 2π^2CS(S^3∖ K) modulo iπ^2ℤ. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.
ER -