On the crossing numbers of a virtual knot
HTML articles powered by AMS MathViewer
- by Shin Satoh and Yumi Tomiyama PDF
- Proc. Amer. Math. Soc. 140 (2012), 367-376 Request permission
Abstract:
We give lower bounds of the real crossing number of a virtual knot in terms of the Jones polynomial and the Miyazawa polynomial. As an application, we prove the existence of a virtual knot such that the real and virtual crossing numbers are equal to $m$ and $n$ for any positive integers $m<n$.References
- Denis Mikhailovich Afanasiev and Vassily Olegovich Manturov, On virtual crossing number estimates for virtual links, J. Knot Theory Ramifications 18 (2009), no. 6, 757–772. MR 2542694, DOI 10.1142/S021821650900718X
- H. A. Dye and Louis H. Kauffman, Virtual crossing number and the arrow polynomial, J. Knot Theory Ramifications 18 (2009), no. 10, 1335–1357. MR 2583800, DOI 10.1142/S0218216509007166
- Roger Fenn, Louis H. Kauffman, and Vassily O. Manturov, Virtual knot theory—unsolved problems, Fund. Math. 188 (2005), 293–323. MR 2191949, DOI 10.4064/fm188-0-13
- J. Green, A Table of Virtual Knots, http://www.math.toronto.edu/drorbn/Students/GreenJ/
- Atsushi Ishii, The pole diagram and the Miyazawa polynomial, Internat. J. Math. 19 (2008), no. 2, 193–207. MR 2384899, DOI 10.1142/S0129167X08004601
- Naoko Kamada, An index of an enhanced state of a virtual link diagram and Miyazawa polynomials, Hiroshima Math. J. 37 (2007), no. 3, 409–429. MR 2376727
- Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
- Louis H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690. MR 1721925, DOI 10.1006/eujc.1999.0314
- Louis H. Kauffman, Detecting virtual knots, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. suppl., 241–282. Dedicated to the memory of Professor M. Pezzana (Italian). MR 1881100
- T. Kishino, On classification of virtual links whose crossing numbers are equal to or less than $6$ (in Japanese), Master Thesis, Osaka City University (2000).
- V. O. Manturov, Teoriya uzlov, in Regular and Chaotic Dynamics, Moscow-Izhevsk, 2005.
- V. O. Manturov, Minimal diagrams of classical and virtual links, arXiv:math/0501393
- Yasuyuki Miyazawa, A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008), no. 11, 1311–1326. MR 2469206, DOI 10.1142/S0218216508006658
- Yasuyuki Miyazawa, A virtual link polynomial and the virtual crossing number, J. Knot Theory Ramifications 18 (2009), no. 5, 605–623. MR 2527679, DOI 10.1142/S0218216509007105
- Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, DOI 10.1016/0040-9383(87)90058-9
- Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309. MR 899051, DOI 10.1016/0040-9383(87)90003-6
- V. G. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, Enseign. Math. (2) 33 (1987), no. 3-4, 203–225. MR 925987
Additional Information
- Shin Satoh
- Affiliation: Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan
- Email: shin@math.kobe-u.ac.jp
- Yumi Tomiyama
- Affiliation: Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan
- Received by editor(s): March 25, 2010
- Received by editor(s) in revised form: November 14, 2010
- Published electronically: May 26, 2011
- Communicated by: Daniel Ruberman
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 367-376
- MSC (2010): Primary 57M25; Secondary 57M27
- DOI: https://doi.org/10.1090/S0002-9939-2011-10917-1
- MathSciNet review: 2833547