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- Title
3-Bridgeness under adding crossings to alternating 3-bridge knots in a 3-bridge representation.
- Authors
Kwon, Bo-Hyun; Kang, Sungmo
- Abstract
In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set E A T k of all essential alternating rational 3-tangles for positive integer k , the authors showed that all knot diagrams in the numerator closure set C N (E A T 2 l + 1) and the denominator closure set C D (E A T 2 l + 2) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set A A T 4 n of additions of alternating rational tangles in E A T 4 . The set A A T 4 n generalizes E A T k and contains it as a subset for some k. We show that any closure set C (A A T 4 n) on A A T 4 n so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in A A T 4 n + 1 is constructed inductively from a tangle diagram in A A T 4 n by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations.
- Subjects
KNOT theory; RECTANGLES; INTEGERS; LOGICAL prediction
- Publication
Journal of Knot Theory & Its Ramifications, 2023, Vol 32, Issue 11, p1
- ISSN
0218-2165
- Publication type
Article
- DOI
10.1142/S0218216523500700