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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Knot theory for self-indexed graphs


Authors: Matías Graña and Vladimir Turaev
Journal: Trans. Amer. Math. Soc. 357 (2005), 535-553
MSC (2000): Primary 57M25, 57M15; Secondary 05C99
Published electronically: August 19, 2004
MathSciNet review: 2095622
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Abstract: We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.


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Additional Information

Matías Graña
Affiliation: Departamento de Matemática - FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, 1428 Buenos Aires, Argentina
Email: matiasg@dm.uba.ar

Vladimir Turaev
Affiliation: IRMA, CNRS - Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg Cedex, France
Email: turaev@math.u-strasbg.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03625-6
PII: S 0002-9947(04)03625-6
Received by editor(s): July 4, 2003
Published electronically: August 19, 2004
Additional Notes: The work of the first author was supported by CONICET (Argentina)
Article copyright: © Copyright 2004 American Mathematical Society