The author discusses 2-adjacency of two-component links and study the relations between the signs of the crossings to realize 2-adjacency and the coefficients of the Conway polynomial of two related links. By discussing the coefficient of the lowest m power in the Homfly polynomial, the author obtains some results and conditions on whether the trivial link is 2-adjacent to a nontrivial link, whether there are two links 2-adjacent to each other, etc. Finally, this paper shows that the Whitehead link is not 2-adjacent to the trivial link, and gives some examples to explain that for any given two-component link, there are infinitely many links 2-adjacent to it. In particular, there are infinitely many links 2-adjacent to it with the same Conway polynomial.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Askitas, N. and Kalfagianni, E., On knot adjacency, Topology and Its Applications, 126, 2002, 63–81.
Bar-Natan, D., The Thistlethwaite link table, http://katlas.math.toronto.edu/wiki/The Thistlethwaite Link Table
Burde, G. and Zieschang, H., Knots, de Gruyter, Berlin, New York, 1985.
Gusarov, M. N., On n-equivalence of knots and invariants of finite degree, Advances in Soviet Math., 18, 1994, 173–192.
Hoste, J., The Arf invariant of a totally proper link, Topology and Its Applications, 18, 1984, 163–177.
Hoste, J., The first coefficient of the Conway polynomial, Proceedings of A. M. S., 95(2), 1985, 299–302.
Howards, H. and Luecke, J., Strongly n-trivial knots, Bull. London Math. Soc., 34, 2002, 431–437.
Kalfagianni, E. and Lin, X. S., Knot adjacency, genus and essential tori, Pacific J. Math., 228(2), 2006, 251–275.
Kauffman, L. H., On Knots, Annals of Mathematics Studies, Vol. 115, Princeton, University Press, Princeton, New Jersey, 1987.
Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 1987, 395–497.
Kawauchi, A., A survey of knot theory, Birkhäuser, Basel, Boston, Berlin, 1996.
Lickorish, W. B. R., The unknotting number of a classical knot, Contemporary Mathematics, 44, 1985, 117–119.
Lickorish, W. B. R. and Millett, K. C., Some evaluations of link polynomials, Commemt. Math. Helv., 61, 1986, 349–359.
Lickorish, W. B. R. and Millett, K. C., A polynomial invariant of oriented links, Topology, 26(1), 1987, 107–141.
Masbaum, G. and Vaintrobw, A., A new matrix tree theorem, http://arxiv.org/pdf/math.CO /0109104.pdf
Murakami, H., On derivatives of the Jones polynomial, Kobe J. Math., 3, 1986, 61–64
Scharlemann, M., Unknotting number one knots are prime, Invent. Math., 82, 1985, 37–55.
Stoimenow, A., On unknotting numbers and knot trivadjacency, Mathematica Scandinavica, 94(2), 2004, 227–248.
Tao, Z. X., On 2-adjacency of classical pretzel knots, Journal of Knot Theory and Its Ramifications, 22(11), 2013, 1350066 (13 pages).
Tao, Z. X., Conway polynomial of 2-adjacent knot, Journal of Zhejiang University, 32(1), 2005, 17–20.
Tao, Z. X., 2-Adjacency between knots, Journal of Knot Theory and Its Ramifications, 24(11), 2015, 1550054.
Tao, Z. X., An Evaluation Property of Jones Polynomial of a Link, Journal of Zhejiang University, 41(5), 2014, 509–511.
Torisu, I., On 2-adjacency relation of two-bridge knots and links, Journal of the Australian Mathematical Society, 84(1), 2008, 139–144.
Torisu, I., On 2-adjacency Relation of Links, Proceedings of the International Workshop on Knot Theory for Scientific Objects, 2007, 277–284.
Tsutsumi, Y., Strongly n-trivial links are boundary links, Tokyo J. Math., 30(2), 2007, 343–350.
This work was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY12A01025).
About this article
Cite this article
Tao, Z. On 2-adjacency between links. Chin. Ann. Math. Ser. B 37, 767–776 (2016). https://doi.org/10.1007/s11401-016-1014-0
- Conway polynomial
- Jones polynomial
- Homfly polynomial
2000 MR Subject Classification