No Access

ON COLORED QUANDLE LONGITUDES AND ITS APPLICATIONS TO TANGLE EMBEDDINGS AND VIRTUAL KNOTS

MACIEJ NIEBRZYDOWSKI

The George Washington University, Washington, DC 20052, USA

https://doi.org/10.1142/S0218216506004932Cited by:4 (Source: Crossref)

Abstract

Given a long knot diagram D and a finite quandle Q, we consider the set of all quandle colorings of D with a fixed color q of its initial arc. Using this set we define the family of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum , taken over all . Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Dedicated to Professor Louis Kauffman on the occasion of his 60th birthday

Keywords:

AMSC: Primary 57M25, Secondary 55M99

References

G. Burde and H. Zieschang , Knots ( De Gruyter , 1985 ) . Google Scholar
S. Carter , S. Kamada and M. Saito , Surfaces in 4-Space ( Springer , 2004 ) . Crossref, Google Scholar
M. Eisermann, Knot colouring polynomials, preprint (2005) . Google Scholar
The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4; 2005 (http://www.gap-system.org) . Google Scholar
M. Goussarov, M. Polyak and O. Viro, Topology 39, 1045 (2000), http://arxiv.org/abs/math.GT/9810073. Crossref, Web of Science, Google Scholar
D. Joyce, J. Pure Appl. Algebra 23, 37 (1982). Crossref, Web of Science, Google Scholar
L. H. Kauffman, European J. Combin. 20, 663 (1999), http://arxiv.org/abs/math.GT/9811028. Crossref, Web of Science, Google Scholar
L. H. Kauffman , Handbook of Knot Theory , eds. W. Menasco and M. Thistlethwaite ( Elsevier , 2005 ) . Crossref, Google Scholar
D. A. Krebes, J. Knot Theory Ramifications 8, 321 (1999), http://arxiv.org/abs/math.GT/9902119. Link, Web of Science, Google Scholar
V. O. Manturov, J. Knot Theory Ramifications 13, 1029 (2004). Link, Web of Science, Google Scholar
J. H. Przytycki, D. S. Silver and S. G. Williams, Math. Proc. Cambridge Phil. Soc. 139, 291 (2005), http://arxiv.org/abs/math.GT/0405465. Crossref, Web of Science, Google Scholar
D. Ruberman, J. Knot Theory Ramifications 9, 523 (2000), http://arxiv.org/abs/math.GT/0001141. Link, Web of Science, Google Scholar
F. Waldhausen, Ann. Math. 87, 56 (1968). Crossref, Web of Science, Google Scholar