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Casson invariant of knots defined by random braids
Abstract
We investigate the expectation E(v2) of the Casson invariant v2 for knots represented by two similar types of braids, the rosette s-braids which are products of k factors ∏1≤ 2i≤ sσ2i± 1∏1≤ 2i+≤ sσ2i+1± 1 and the quasitoric s-braids which are products of k factors σ1± 1σ2± 1...σs−1± 1. If k is of the form ns+1, E(v2)/c∼ (s+1)/24 where c is the number of crossing points of braid; if k is of the form ns−1, E(v2)/c∼ (5−s)/24.\\ If we go through all $3$-braids, regardless of the shape of their diagram, we get E(v2)/c∼ 1/8. \\ For knots with a checkerboard diagram, E(v2)/c∼ 1/16.
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BibTeX
@misc{ville2026casson,
title = {{Casson invariant of knots defined by random braids}},
author = {Marina Ville},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01188},
eprint = {2611.01188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01188},
}
RIS
TY - UNPB
TI - Casson invariant of knots defined by random braids
AU - Marina Ville
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01188
UR - https://arxiv.org/abs/2611.01188
AB - We investigate the expectation E(v_2) of the Casson invariant v_2 for knots represented by two similar types of braids, the rosette s-braids which are products of k factors ∏_1≤ 2i≤ sσ_2i^± 1∏_1≤ 2i+≤ sσ_2i+1^± 1 and the quasitoric s-braids which are products of k factors σ_1^± 1σ_2^± 1...σ_s−1^± 1. If k is of the form ns+1, E(v_2)/c∼ (s+1)/24 where c is the number of crossing points of braid; if k is of the form ns−1, E(v_2)/c∼ (5−s)/24.\\ If we go through all $3$-braids, regardless of the shape of their diagram, we get E(v_2)/c∼ 1/8. \\ For knots with a checkerboard diagram, E(v_2)/c∼ 1/16.
ER -