TY - JOUR

T1 - Euclidean volumes of hyperbolic knots

AU - Abrosimov, Nikolay

AU - Kolpakov, Alexander

AU - Mednykh, Alexander

N1 - Received by the editors September 4, 2021, and, in revised form, August 22, 2022, and November 18, 2022. 2020 Mathematics Subject Classification. Primary 57K10, 57M50, 11R04. The first and third authors were supported by the state contract of Sobolev Institute of Mathematics (project no. FWNF-2022-0005). The second author was supported by the Swiss National Science Foundation (projects PP00P2–170560 and PP00P2–202667).

PY - 2024/2/1

Y1 - 2024/2/1

N2 - The hyperbolic structure on a 3 3 –dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.

AB - The hyperbolic structure on a 3 3 –dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85181951358&origin=inward&txGid=ba4c84da532144f011d6120be3c1e9df

UR - https://www.mendeley.com/catalogue/cd58b304-cdef-3f40-81f9-8bc36a1f4d27/

U2 - 10.1090/proc/16353

DO - 10.1090/proc/16353

M3 - Article

VL - 152

SP - 869

EP - 881

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -