Integral face-edge duality for knot triangulations
Abstract
For an ordered ideal triangulation of a knot exterior in S3, we give a cochain construction relating its face matrix C to its preferred-longitude gluing matrices A,B. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of kerℤB and an exact sequence from the two face cokernels to the gluing cokernel, with quotient ℤ/2ℤ. In the unhalved longitude convention this implies |det B|=2|det C|2. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen--Kashaev volume conjecture remain open here.
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BibTeX
@misc{kambampati2026integral,
title = {{Integral face-edge duality for knot triangulations}},
author = {Sai Kambampati},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04103},
eprint = {2610.04103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04103},
}
RIS
TY - UNPB
TI - Integral face-edge duality for knot triangulations
AU - Sai Kambampati
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04103
UR - https://arxiv.org/abs/2610.04103
AB - For an ordered ideal triangulation of a knot exterior in S^3, we give a cochain construction relating its face matrix C to its preferred-longitude gluing matrices A,B. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of ker_ℤB and an exact sequence from the two face cokernels to the gluing cokernel, with quotient ℤ/2ℤ. In the unhalved longitude convention this implies |det B|=2|det C|^2. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen--Kashaev volume conjecture remain open here.
ER -