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

DOI:
https://doi.org/10.1090/S0002-9947-04-03625-6

Published electronically:
August 19, 2004

MathSciNet review:
2095622

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-04-03625-6

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