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Knot adjacency from a surgical view of Alexander invariants

Yasutaka Nakanishi

Department of Mathematics, Graduate School of Science, Kobe University, Rokko, Nada-ku, Kobe 657-8501, Japan

E-mail Address: nakanisi@math.kobe-u.ac.jp

Corresponding author.

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and

Masahiro Shimoda

Department of Mathematics, Graduate School of Science, Kobe University, Rokko, Nada-ku, Kobe 657-8501, Japan

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https://doi.org/10.1142/S0218216517500985Cited by:0 (Source: Crossref)

Abstract

For two knots $K_{1}$ $K_{1}$ and $K_{2}$ $K_{2}$ , $K_{1}$ $K_{1}$ is said to be $n$ $n$ -adjacent to $K_{2}$ $K_{2}$ , if $K_{1}$ $K_{1}$ admits a knot diagram containing $n$ $n$ crossings such that crossing changes at any non-empty subset of them yield a knot diagram of $K_{2}$ $K_{2}$ . Using a surgical view of Alexander invariants, we will characterize the Alexander polynomials of knots which are $2$ $2$ -adjacent to the trivial knot.

Keywords:

AMSC: 57M25

References

1. N. Askitas and E. Kalfagianni, On knot adjacency, Topology Appl. 126 (2002) 63–81. Crossref, Web of Science, Google Scholar
2. S. Fukaguchi, Spatial embeddings of $K_{3, 3}$ $K_{3, 3}$ graph and $2$ $2$ -adjacency of knots, Master Thesis, Saitama University (2017). Google Scholar
3. M. Gusarov, On $n$ $n$ -equivalence of knots and invariants of finite degree, in Topology of manifolds and varieties, Advances in Siviet Mathematics, Vol. 18 (Amrican Mathematical Society, Providence, R.I., 1994), pp. 173–192. Crossref, Google Scholar
4. H. Howards and J. Luecke, Strongly $n$ $n$ -trivial knots, Bull. London Math. Soc. 34 (2002) 431–437. Crossref, Web of Science, Google Scholar
5. E. Kalfagianni, Surgery $n$ $n$ -triviality and companion tori, J. Reine Angew. Math. 669 (2012) 151–164. Web of Science, Google Scholar
6. E. Kalfagianni, Cosmetic crossing changes of fibered knotsi, J. Knot Theory Ramifications 13 (2004) 441–456. Link, Web of Science, Google Scholar
7. E. Kalfagianni and X.-S. Lin, Knot adjacency and satellites, Topology Appl. 138 (2004) 207–217. Crossref, Web of Science, Google Scholar
8. E. Kalfagianni and X.-S. Lin, Knot adjacency, genus and essential tori, Pacific J. Math. 228 (2006) 251–276. Crossref, Web of Science, Google Scholar
9. T. Kanenobu and H. Murakami, Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986) 499–502. Crossref, Web of Science, Google Scholar
10. J. Levine, A characterization of knot polynomials, Topology 4 (1965) 135–141. Crossref, Google Scholar
11. Y. Nakanishi, Alexander polynomials of knots which are transformed into the trefoil knots by a single crossing change, Kyungpook Math. J. 52 (2012) 201–208. Crossref, Google Scholar
12. Y. Ohyama, On the triviality index of knots, Topology Appl. 37 (1990) 249–255. Crossref, Web of Science, Google Scholar
13. D. Rolfsen, A surgical view of Alexander's polynomial, in Geometric Topology, Lecture Notes in Mathematics, Vol. 438 (Springer-Verlag, Berlin and New York, 1974), pp. 415–423. Google Scholar
14. M. Shimoda, Musubime no rinsetsusei to soujisei nitsuite (in Japanese) [Adjacency and similarity of knots], Master Thesis, Kobe University (2016). Google Scholar
15. K. Taniyama, On similarity of links, Gakujutu Kenkyu (issued by the school of education of Waseda University) 14 (1992) 33–36. Google Scholar
16. Z.-X. Tao, On $2$ $2$ -adjacency of classical pretzel knots, J. Knot Theory Ramification 22 (2013) 1350066. Link, Web of Science, Google Scholar
17. Z.-X. Tao, On knot $2$ $2$ -adjacency, Topology Appl. 172 (2014) 72–78. Crossref, Google Scholar
18. Z.-X. Tao, Adjacency between knots, J. Knot Theory Ramification 24 (2015) 1550054. Link, Web of Science, Google Scholar
19. Z.-X. Tao, $2$ $2$ -adjacency between pretzel links and the trivial link, Topology Appl. 214 (2016) 186–191. Crossref, Google Scholar
20. I. Torisu, On strongly $n$ $n$ -trivial $2$ $2$ -bridge knots, Math. Proc. Camb. Phil. Soc. 137 (2004) 613–616. Crossref, Web of Science, Google Scholar
21. I. Torisu, A note on strongly $n$ $n$ -trivial links, J. Knot Theory Ramification 14 (2005) 565–569. Link, Web of Science, Google Scholar
22. I. Torisu, Two-bridge links with strong triviality, Tokyo J. Math. 29 (2006) 233–237. Crossref, Google Scholar
23. I. Torisu, On strongly $1$ $1$ -trivial Montesinos knot, Hiroshima Math. J. 38 (2008) 225–229. Crossref, Web of Science, Google Scholar
24. I. Torisu, On $2$ $2$ -adjacency relation of two-bridge knots and links, J. Austr. Math. Soc. 84 (2008) 139–144. Crossref, Web of Science, Google Scholar
25. I. Torisu, $3$ $3$ -adjacency relation of two-bridge knots and links, J. Knot Theory Ramifications 21 (2012) 250051. Link, Web of Science, Google Scholar
26. I. Torisu, A remark on $2$ $2$ -adjacency relation between Montesinos knots and $2$ $2$ -bridge knots, Topology Appl. 198 (2016) 34–37. Crossref, Google Scholar
27. Y. Tsutsumi, Strongly $n$ $n$ -trivial links are boundary links, Tokyo J. Math. 30 (2007) 343–350. Crossref, Google Scholar
28. H. Wendt, Die Gordische Auflösung von Knoten, Math. Z. 42 (1937) 680–696. Crossref, Google Scholar