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The braid indices of most links remain unknown as there is no known universal method for determining the braid index of an arbitrary knot. This is also the case for alternating knots. We show that if K is an alternating knot, then the braid index of any reverse parallel link of K can be precisely determined. Specifically, if D is a reduced diagram of K, v+(D) (resp. v−(D)) is the number of regions in the checkerboard shading of D for which all crossings are positive (resp. negative) and w(D) is the writhe of D, then the braid index of a reverse parallel link of K with framing f, denoted by 𝕂f, is given by the precise formula
where a(D) = −v−(D) + w(D) and b(D) = v+(D) + w(D).
Diao, Yuanan 1 ; Morton, Hugh 2
@article{10_2140_agt_2024_24_2957,
author = {Diao, Yuanan and Morton, Hugh},
title = {The braid indices of the reverse parallel links of alternating knots},
journal = {Algebraic and Geometric Topology},
pages = {2957--2970},
publisher = {mathdoc},
volume = {24},
number = {5},
year = {2024},
doi = {10.2140/agt.2024.24.2957},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2957/}
}
TY - JOUR AU - Diao, Yuanan AU - Morton, Hugh TI - The braid indices of the reverse parallel links of alternating knots JO - Algebraic and Geometric Topology PY - 2024 SP - 2957 EP - 2970 VL - 24 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2957/ DO - 10.2140/agt.2024.24.2957 ID - 10_2140_agt_2024_24_2957 ER -
%0 Journal Article %A Diao, Yuanan %A Morton, Hugh %T The braid indices of the reverse parallel links of alternating knots %J Algebraic and Geometric Topology %D 2024 %P 2957-2970 %V 24 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2957/ %R 10.2140/agt.2024.24.2957 %F 10_2140_agt_2024_24_2957
Diao, Yuanan; Morton, Hugh. The braid indices of the reverse parallel links of alternating knots. Algebraic and Geometric Topology, Tome 24 (2024) no. 5, pp. 2957-2970. doi: 10.2140/agt.2024.24.2957
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