No Access

TWIN GROUPS OF VIRTUAL 2-BRIDGE KNOTS AND ALMOST CLASSICAL KNOTS

TAKUJI NAKAMURA

Department of Engineering Science, Osaka Electro-Communication University, Hatsu-cho 18-8, Neyagawa, 572-8530, Japan

Search for more papers by this author

YASUTAKA NAKANISHI

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan

Search for more papers by this author

SHIN SATOH

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan

Search for more papers by this author

, and

YUMI TOMIYAMA

Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-0013, Japan

Search for more papers by this author

https://doi.org/10.1142/S0218216512500952Cited by:2 (Source: Crossref)

Abstract

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups.

Keywords:

AMSC: Primary 57M25, Secondary 57Q35

References

E. Byberi and V. V. Chernov, Commun. Contemp. Math. 10(1), 1013 (2008). Link, Google Scholar
H. A. Dye and L. H. Kauffman, J. Knot Theory Ramifications 18(10), 1335 (2009). Link, Web of Science, Google Scholar
R. Fenn, R. Rimányi and C. Rourke, Topology 36(1), 123 (1997). Crossref, Web of Science, Google Scholar
M. Goussarov, M. Polyak and O. Viro, Topology 39(5), 1045 (2000). Crossref, Web of Science, Google Scholar
J. Green, A table of virtual knots, http://www.math.toronto.edu/~drorbn/Students/GreenJ/ . Google Scholar
A. Henrich, J. Knot Theory Ramifications 19(4), 461 (2010). Link, Web of Science, Google Scholar
M. Hirasawa, N. Kamada and S. Kamada, J. Knot Theory Ramifications 20(6), 881 (2031). Link, Web of Science, Google Scholar
A. Ichimori and T. Kanenobu, Ribbon torus knots presented by virtual knots with up to 4 crossings, preprint (2009) . Google Scholar
T. Imabeppu, On normalized arrow polynomials of checkerboard colorable virtual knots, preprint . Google Scholar
A. Ishii, Int. J. Math. 19, 193 (2008). Link, Web of Science, Google Scholar
N. Kamada, Osaka J. Math. 39(2), 325 (2002). Web of Science, Google Scholar
N. Kamada and S. Kamada, J. Knot Theory Ramifications 9(1), 93 (2000). Link, Web of Science, Google Scholar
T. Kanenobu, J. Knot Theory Ramifications 10(1), 89 (2001). Link, Web of Science, Google Scholar
L. H. Kauffman, European J. Combin. 20(7), 663 (1999). Crossref, Web of Science, Google Scholar
Y. Miyazawa, J. Knot Theory Ramifications 17(11), 1311 (2008). Link, Web of Science, Google Scholar
S. Nelson, J. Knot Theory Ramifications 10(6), 931 (2001). Link, Web of Science, Google Scholar
S. Satoh and Y. Tomiyama, Proc. Amer. Math. Soc. 140(1), 367 (2012). Crossref, Web of Science, Google Scholar
D. S. Silver and S. G. Williams, Knots in Hellas '98 24 (World Scientific Publications, River Edge, NJ, 2000) pp. 440–451. Link, Google Scholar
D. S. Silver and S. G. Williams, J. Knot Theory Ramifications 15(8), 1079 (2006). Link, Web of Science, Google Scholar
Y. Tomiyama, Kasou-musubime no kouten-suu to suteito-suu (in Japanese), Crossing numbers and state number of virtual knot, Master Thesis, Kobe University (2010) . Google Scholar