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Interlace polynomials

✍ Scribed by Martin Aigner; Hein van der Holst

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
275 KB
Volume
377
Category
Article
ISSN
0024-3795

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✦ Synopsis

In a recent paper Arratia, BollobΓ‘s and Sorkin discuss a graph polynomial defined recursively, which they call the interlace polynomial q(G, x). They present several interesting results with applications to the Alexander polynomial and state the conjecture that |q(G, -1)| is always a power of 2. In this paper we use a matrix approach to study q(G, x). We derive evaluations of q(G, x) for various x, which are difficult to obtain (if at all) by the defining recursion. Among other results we prove the conjecture for x = -1. A related interlace polynomial Q(G, x) is introduced. Finally, we show how these polynomials arise as the Martin polynomials of a certain isotropic system as introduced by Bouchet.

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We show that two classical theorems in graph theory and a simple result concerning the interlace polynomial imply that if K is a reduced alternating link diagram with n β‰₯ 2 crossings, then the determinant of K is at least n. This gives a particularly simple proof of the fact that reduced alternating

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In this paper we give an explicit formula for the interlace polynomial at x = -1 for any graph, and as a result prove a conjecture of Arratia et al. that states that it is always of the form Β±2 s . We also give a description of the graphs for which s is maximal.