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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Concordance crosscap numbers of knots and the Alexander polynomial


Author: Charles Livingston
Journal: Proc. Amer. Math. Soc. 136 (2008), 3351-3353
MSC (2000): Primary 57M25
Published electronically: April 25, 2008
MathSciNet review: 2407102
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Abstract: For a knot $ K$ the concordance crosscap number, $ c(K)$, is the minimum crosscap number among all knots concordant to $ K$. Building on work of G. Zhang, which studied the determinants of knots with $ c(K) < 2$, we apply the Alexander polynomial to construct new algebraic obstructions to $ c(K) < 2$. With the exception of low crossing number knots previously known to have $ c(K) < 2$, the obstruction applies to all but four prime knots of 11 or fewer crossings.


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Additional Information

Charles Livingston
Affiliation: Department of Mathematics, Indiana University, 123 Rawles Hall, Bloomington, Indiana 47405

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09481-1
PII: S 0002-9939(08)09481-1
Received by editor(s): May 11, 2007
Published electronically: April 25, 2008
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.