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Algebraic Gordian distance

Jie Chen

Department of Mathematics & Statistics, McMaster University, 1280 Main Street West Hamilton, Ontario, Canada L8S 4K1

https://doi.org/10.1142/S021821651950024XCited by:1 (Source: Crossref)

Abstract

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

Keywords:

AMSC: 57M25, 57M27

References

1. D. Bar-Natan, Knot Atlas (2019), http://katlas.org/wiki/Main_Page. Google Scholar
2. R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. Math. 65(2) (1957) 340–356. Crossref, Web of Science, Google Scholar
3. M. Borodzik and S. Friedl, Knotorious (2017), https://www.mimuw.edu.pl/˜mcboro/knotorious.php. Google Scholar
4. M. E. Fogel, Knots with algebraic unknotting number one, Pacific J. Math. 163 (1994) 277–295. Crossref, Web of Science, Google Scholar
5. S. Friedl and M. Powell, A calculation of Blanchfield pairings of 3-manifolds and knots, Mosc. Math. J. 17(1) (2017) 59–77. Crossref, Web of Science, Google Scholar
6. K. Ichihara and I. D. Jong, Gromov hyperbolicity and a variation of the Gordian complex, Proc. Japan Acad. Ser. A Math. Sci. 87(2) (2011) 17–21. Crossref, Google Scholar
7. I. D. Jong, Alexander polynomials of alternating knots of genus two, Osaka J. Math. 46(2) (2009) 353–371. Web of Science, Google Scholar
8. I. D. Jong, Alexander polynomials of alternating knots of genus two II, J. Knot Theory Ramifications 19(8) (2010) 1075–1092. Link, Web of Science, Google Scholar
9. A. Kawauchi, On the Alexander polynomials of knots with Gordian distance one, Topol. Appl. 159(4) (2012) 948–958. Crossref, Web of Science, Google Scholar
10. H. Kondo, Knots of unknotting number 1 and their Alexander polynomials, Osaka J. Math. 16(2) (1979) 551–559. Google Scholar
11. J. Levine, The role of the Seifert matrix in knot theory, in Actes du Cong. Int. des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris (1971), pp. 95–98. Google Scholar
12. W. B. R. Lickorish, The unknotting number of a classical knot, Contemp. Math. 44 (1985) 117–121. Crossref, Google Scholar
13. H. Moon, Calculating knot distances and solving tangle equations involving montesinos links, Ph.D. thesis, The University of Iowa (2010). Google Scholar
14. H. Murakami, Some metrics on classical knots, Math. Ann. 270(1) (1985) 35–45 Crossref, Web of Science, Google Scholar
15. H. Murakami, Algebraic unknotting operation, Questions Answers Gen. Topology 8(1) (1990) 283–292. Google Scholar
16. K. Murasugi, On the signature of links, Topology 9 (1970) 283–298. Crossref, Google Scholar
17. Y. Nakanishi, Local moves and Gordian complexes. II, Kyungpook Math. J. 47(3) (2007) 329–334. Google Scholar
18. D. Rolfsen, Knots and Links, 2nd edn. Mathematics Lecture Series (Publish or Perish, Berkeley, California, 1990). Google Scholar
19. O. Saeki, On algebraic unknotting numbers of knots, Tokyo J. Math. 22(2) (1999) 425–443. Crossref, Google Scholar
20. T. Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ. 5(3) (1977) 451–456. Google Scholar
21. H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110(1) (1935) 571–592. Crossref, Google Scholar
22. H. Seifert, Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen, Abh. Math. Sem. Univ. Hamburg 11(1) (1935) 84–101. Crossref, Google Scholar
23. H. F. Trotter, Homology of group systems with applications to knot theory, Ann. Math. 76(2) (1962) 464–498. Crossref, Web of Science, Google Scholar
24. H. F. Trotter, On $S$ -equivalence of Seifert matrices, Invent. Math. 20 (1973) 173–207. Crossref, Web of Science, Google Scholar