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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Circuit partitions and the homfly polynomial of closed braids

Author: François Jaeger
Journal: Trans. Amer. Math. Soc. 323 (1991), 449-463
MSC: Primary 57M25; Secondary 57M15
MathSciNet review: 986693
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Abstract: We present an expansion of the homfly polynomial $ P(D,z,a)$ of a braid diagram $ D$ in terms of its circuit partitions. Another aspect of this result is an expression of $ P(D,z,a)$ as the trace of a matrix associated to $ D$ in a simple way. We show how certain degree properties of the homfly polynomial can be derived easily from this model. In particular we obtain that if $ D$ is a positive braid diagram on $ n$ strings with $ w$ crossings, the maximum degree of $ P(D,z,a)$ in the variable $ a$ equals $ n - 1 - w$.

Nous présentons une expansion pour le polynôme homfly $ P(D,z,a)$ d'un diagramme de tresse $ D$ en termes de ses partitions en circuits. Un autre aspect de ce résultat consiste en une expression de $ P(D,z,a)$ comme trace d'une matrice associee de façon simple à $ D$. Nous montrons comment certaines propriétés de degré du polynôme homfly dérivent simplement de ce modèle. En particulier nous obtenons que pour un diagramme de tresse positif $ D$ à $ n$ brins et $ w$ croisements, le degré maximum de $ P(D,z,a)$ en la variable $ a$ est égal à $ n - 1- w$.

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Keywords: Links, braids, diagrams, graphs, polynomial invariants, combinatorial models, transfer matrices
Article copyright: © Copyright 1991 American Mathematical Society