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3-Bridgeness under adding crossings to alternating 3-bridge knots in a 3-bridge representation.
- Source :
-
Journal of Knot Theory & Its Ramifications . Oct2023, Vol. 32 Issue 11, p1-22. 22p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *KNOT theory
*RECTANGLES
*INTEGERS
*LOGICAL prediction
Subjects
Details
- Language :
- English
- ISSN :
- 02182165
- Volume :
- 32
- Issue :
- 11
- Database :
- Academic Search Index
- Journal :
- Journal of Knot Theory & Its Ramifications
- Publication Type :
- Academic Journal
- Accession number :
- 174525427
- Full Text :
- https://doi.org/10.1142/S0218216523500700