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The Wriggle polynomial for virtual tangles

Oxford College of Emory University, 801 Emory Street, Oxford, GA 30054, USA

https://doi.org/10.1142/S0218216519500871Cited by:0 (Source: Crossref)

Abstract

We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalizations (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.

Keywords:

AMSC: 57M27, 57M99

References

1. H. Dye , Smoothed Invariants, J. Knot Theory Ramifications 21(13) (2012) 1240003. Link, Web of Science, Google Scholar
2. L. C. Folwaczny and L. H. Kauffman , A linking number definition of the affine index polynomial, J. Knot Theory Ramifications 22(12) (2013) 1341004. Link, Web of Science, Google Scholar
3. M. Goussarov, M. Polyak and O. Viro , Finite-type invariants of classical and virtual knots, Topology 39(5) (2000) 1045–1068. Crossref, Web of Science, Google Scholar
4. A. Henrich , A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19(4) (2010) 461–487. Link, Web of Science, Google Scholar
5. D. P. Ilyutko, V. O. Manturov and I. M. Nikonov , Virtual knot invariants arising from parities, in Knots in Poland III Part 1, Vol. 100 (Banach Center Publications, 2014), pp. 99–130. Crossref, Google Scholar
6. L. H. Kauffman , Virtual knot theory, Eur. J. Combin. 20(7) (1999) 663–690. Crossref, Web of Science, Google Scholar
7. L. H. Kauffman , An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013) 1340007. Link, Web of Science, Google Scholar
8. L. H. Kauffman , Virtual knot cobordism and the affine index polynomial, J. Knot Theory Ramifications 27(11) (2018) 1843017. Link, Web of Science, Google Scholar
9. N. Petit , Index polynomials for virtual tangles, J. Knot Theory Ramifications 27(12) (2018) 1850073. Link, Web of Science, Google Scholar
10. N. Petit , Finite-type invariants of long and framed virtual knots, J. Knot Theory Ramifications 28(10) (2019) 1950064. Link, Web of Science, Google Scholar