2023-09-26T09:16:20Z
https://oa.upm.es/cgi/oai2
oai:oa.upm.es:14153
2016-04-21T13:40:51Z
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A geometric characterization of the upper bound for the span of the jones polynomial
Gonzalez Meneses, Juan
Gonzalez Manchon, Pedro Maria
Industrial Engineering
Mathematics
Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram
E.U.I.T. Industrial (UPM)
https://creativecommons.org/licenses/by-nc-nd/3.0/es/
2011-07
info:eu-repo/semantics/article
Article
Journal of Knot Theory and Its Ramifications, ISSN 0218-2165, 2011-07, Vol. 20, No. 7
PeerReviewed
application/pdf
spa
http://www.worldscientific.com/doi/abs/10.1142/S0218216511009005?journalCode=jktr
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/altIdentifier/doi/10.1142/S0218216511009005
https://oa.upm.es/14153/