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Quadruple crossing number of knots and links

Published online by Cambridge University Press:  20 November 2013

COLIN ADAMS
COLIN ADAMS*
Affiliation:
Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, U.S.A. e-mail: [email protected]
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Abstract

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number c4(K). In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 241 - 253
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Adams, C.Triple crossing number of knots and links. J. Knot Theory Ramifications 22 (2013), 1350006.Google Scholar
[2]Adams, C., Crawford, T., DeMeo, B., Landry, M., Lin, A., Montee, M., Park, S., Venkatesh, S. and Yhee, F. Knot projections with a single multi-crossing. ArXiv:1208.5742 2012.Google Scholar
[3]Adams, C., Capovilla–Searle, O., Freeman, J., Irvine, D., Petti, S., Vitek, D., Weber, A. and Zhang, S. Multicrossing numbers and the span of the bracket polynomial, ArXiv 1311.0526 2013.Google Scholar
[4]Adams, C., Capovilla–Searle, O., Freeman, J., Irvine, D., Petti, S., Vitek, D., Weber, A. and Zhang, S. Bounds on Übercrossing and petal numbers for knots, in preparation 2013.Google Scholar
[5]Jones, V. F. R.Hecke algebra representations of braid groups and link polynomials. Ann. of Math. 126 (1987), 335–388.Google Scholar
[6]Kauffman, L.New invariants in the theory of knots. Amer. Math. Monthly 95 (1988), 195–242.Google Scholar
[7]Murasugi, K.Jones polynomials of alternating links. Trans. Amer. Math. Soc. 295 (1986), 147–174.Google Scholar
[8]Pach, J. and G. Toh Degenerating crossing numbers. Discrete Comput. Geom. 41 (2009), 376–384.CrossRefGoogle Scholar
[9]Tanaka, H. and Teragaito, M. Triple crossing numbers of graphs, ArXiv:1002.4231 (2010).Google Scholar
[10]Thistlethwaite, M.A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297–309.Google Scholar