The Braid Indices of Pretzel Links: A Comprehensive Study, Part II
arxiv(2024)
摘要
This paper is the second part of our comprehensive study on the braid index
problem of pretzel links. Our ultimate goal is to completely determine the
braid indices of all pretzel links, alternating or non alternating. In our
approach, we divide the pretzel links into three types as follows. Let D be a
standard diagram of an oriented pretzel link ℒ, S(D) be the
Seifert circle decomposition of D, and C_1, C_2 be the Seifert circles in
S(D) containing the top and bottom long strands of D respectively, then
ℒ is classified as a Type 1 (Type 2) pretzel link if C_1≠C_2
and C_1, C_2 have different (identical) orientations. In the case that
C_1=C_2, then ℒ is classified as a Type 3 pretzel link. In our
previous paper, we succeeded in reaching our goal for all Type 1 and Type 2
pretzel links. That is, we successfully derived precise braid index formulas
for all Type 1 and Type 2 pretzel links. In this paper, we present the results
of our study on Type 3 pretzel links. In this case, we are very close to
reaching our goal. More precisely, with the exception of a small percentage of
Type 3 pretzel links, we are able to determine the precise braid indices for
the majority of Type 3 pretzel links. Even for those exceptional ones, we are
able to determine their braid indices within two consecutive integers. With
some numerical evidence, we conjecture that in such a case, the braid index of
the Type 3 pretzel link is given by the larger of the two consecutive integers
given by our formulas.
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