Research ArticlesNo Access

New parities and coverings over free knots

Vassily Olegovich Manturov

Bauman Moscow State Technical University and Chelyabinsk State University, Russia

https://doi.org/10.1142/S0218216516500772Cited by:0 (Source: Crossref)

Abstract

In the present paper, we develop the parity theory invented in [V. O. Manturov, Mathematics201(5) (2010) 693733]; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, where parities were previously applied.

Keywords:

AMSC: 57M25, 57M27

References

1. M. W. Chrisman, Knots in virtually fibered 3-manifolds, arXiv e-prints, May 2014. Google Scholar
2. M. W. Chrisman and V. O. Manturov, Fibered knots and virtual knots, J. Knot Theory Ramifications 22(12) (2013) 1341003, 23. Link, Web of Science, Google Scholar
3. P. R. Cromwell and H. R. Morton, Distinguishing mutants by knot polynomials, J. Knot Theory Ramifications 5(2) (1996) 225–238. Link, Web of Science, Google Scholar
4. N. Habegger and X.-S. Lin, The classification of links up to homotopy, J. Am. Math. Soc. 2 (American Mathematical Society) 3(2) (1990) 389–419. Crossref, Google Scholar
5. D. P. Ilyutko, V. O. Manturov and I. M. Nikonov, Parity in knot theory and graph-links, J. Math. Sci. 193(6) (2013) 809–965. Crossref, Google Scholar
6. L. H. Kauffman, Virtual knot theory, Eur. J. Combinatorics 20(7) (1999) 663–690. Crossref, Web of Science, Google Scholar
7. L. H. Kauffman, A self-linking invariant of virtual knots, Fundamenta Mathematicae 184 (2004) 135–158. Crossref, Web of Science, Google Scholar
8. V. A. Krasnov and V. O. Manturov, Graph-valued invariants of virtual and classical links and minimality problem, J. Knot Theory Ramifications 22(12) (2013) 1341006, 14. Link, Web of Science, Google Scholar
9. V. O. Manturov and D. P. Ilyutko, Virtual Knots: The State of The Art, Series on Knots and Everything, Vol. 51 (World Scientific, Singapore, 2013). Google Scholar
10. V. O. Manturov, The Khovanov complex for virtual links, J. Math. Sci. (New York) 144(5) (2007) 4451–4467. Crossref, Google Scholar
11. V. O. Manturov, Parity in knot theory, Sbornik: Mathematics 201(5) (2010) 693–733. Crossref, Web of Science, Google Scholar
12. V. O. Manturov, Parity and cobordisms of free knots, Sbornik: Mathematics 203(2) (2012) 196–223. Crossref, Web of Science, Google Scholar
13. J. Milnor, Link groups, Ann. Math. 59 (1954) 177–195. Crossref, Web of Science, Google Scholar
14. V. G. Turaev, Topology of words, Proc. Lond. Math. Soc. 95(3) (2007) 360–412. Crossref, Web of Science, Google Scholar