A geometric characterization of the upper bound for the span of the Jones polynomial
Let D be a link diagram with n crossings, sA and sB its extreme states and |sAD| (resp. |sBD|) the number of simple closed curves that appear when smoothing D according to sA (resp. sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizi...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2011 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/42282 |
| Acceso en línea: | http://hdl.handle.net/11441/42282 https://doi.org/10.1142/S0218216511009005 |
| Access Level: | acceso abierto |
| Palabra clave: | K-almost alternating diagram Circle number Surgery Dealternator connected diagram Dealternator reduced diagram Jones polynomial Span |
| Sumario: | Let D be a link diagram with n crossings, sA and sB its extreme states and |sAD| (resp. |sBD|) the number of simple closed curves that appear when smoothing D according to sA (resp. 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. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al, 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. |
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