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Multi-crossing number for knots and the Kauffman bracket polynomial

Published online by Cambridge University Press:  02 November 2016

COLIN ADAMS ,
ORSOLA CAPOVILLA-SEARLE ,
JESSE FREEMAN ,
DANIEL IRVINE ,
SAMANTHA PETTI ,
DANIEL VITEK ,
ASHLEY WEBER  and
SICONG ZHANG
COLIN ADAMS
Affiliation:
Department of Mathematics, Williams College, Williamstown, MA 01267, U.S.A. e-mail: [email protected]
ORSOLA CAPOVILLA-SEARLE
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, U.S.A. e-mail: [email protected]
JESSE FREEMAN
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307, U.S.A. e-mail: [email protected]
DANIEL IRVINE
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, U.S.A. e-mail: [email protected]
SAMANTHA PETTI
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, U.S.A. e-mail: [email protected]
DANIEL VITEK
Affiliation:
Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544-1000, U.S.A. e-mail: [email protected]
ASHLEY WEBER
Affiliation:
Department of Mathematics, 151 Thayer Street, Brown University, Providence, RI 02912, U.S.A. e-mail: [email protected]
SICONG ZHANG
Affiliation:
Department of Mathematics, Building 380, Stanford University, Stanford, CA 94305, U.S.A. e-mail: [email protected]
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Abstract

A multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3:

$$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$
We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 147 - 178
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Adams, C. Triple crossing number of knots and links. J. Knot Theory Ramifiaction 22 (02) (2013).Google Scholar
[2] Adams, C. Quadruple crossing number of knots and links. Math. Proc. Camb. Phil. Soc. 156 (2) (2014), 241–253.CrossRefGoogle Scholar
[3] 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. J. Knot Theory Ramifiaction 24 (2) (2015).Google Scholar
[4] Adams, C., Crawford, T., DeMeo, B., Landry, M., Lin, A. T., Park, S., Montee, M., Venkatesh, S. and Yhee, F. Knot projections with a single multi-crossing. J. Knot Theory Ramification 24 (3) (2015).Google Scholar
[5] Culler, M., Dunfield, N., Goerner, M. and Weeks, J. SnapPy, a computer program for studying the topology of 3-manifolds. Available at url http://snappy.computop.org (8/30/16), 2016.Google Scholar
[6] Kauffman, L. State models and the jones polynomial. Topology 26 (3) (1987), 395–407.Google Scholar
[7] Kauffman, L. New invariants in the theory of knots. Amer. Math. Monthly 95 (3) (1988), 195–242.Google Scholar
[8] Murasugi, K. Jones polynomial and classical conjectures in knot theory. Topology 26 (1987), 187–194.Google Scholar
[9] Pach, J. and Toth, G. Degenerate crossing numbers. Discrete Comput. Geom. 41 (2009), 376–384.Google Scholar
[10] Tanaka, H. and Teragaito, M. Triple crossing numbers of graphs. ArXiv:1002.4231 (2010).Google Scholar
[11] Thistlethwaite, M. A spanning tree expansion of the jones polynomial. Topology 26 (3) (1987), 297–309.Google Scholar