Download e-book for iPad: An Index of a Graph With Applications to Knot Theory by Kunio Murasugi

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By Kunio Murasugi

ISBN-10: 0821825704

ISBN-13: 9780821825709

This publication provides a impressive software of graph thought to knot thought. In knot conception, there are many simply outlined geometric invariants which are tremendous tricky to compute; the braid index of a knot or hyperlink is one instance. The authors assessment the braid index for lots of knots and hyperlinks utilizing the generalized Jones polynomial and the index of a graph, a brand new invariant brought right here. This invariant, that is decided algorithmically, might be of specific curiosity to computing device scientists.

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See Fig. 6. Fig. 13, putting p' = gi, p" 46 KUNIO MURASUGI AND J O Z EF H. 2, q = Pi and q' = p2 . Therefore, «0 + ( I 3 ) (^) = 0 . complete. 11 is now • §10 Braid i n d e x of special alternatin g links Since an alternating link is a * -product of special alternating links, it is natural to expect that the braid index of an alternating link is completely determined by those of its *-components. ) Although this is not proved yet, the determination of the braid index of a special alternating link will be the first step toward the complete determination of the braid index of an alternating link.

D), = T/>+(D°). + n(D) - n+(D) = j>+{D). Z}° contributes a maximal term ^ + (D)(^) • On the other hand, the maximal term in CL^+{D){Z) 2:^+(D) in Dl can contribute is 40 KUNIO MURASUGI AND J O Z E F H. PRZYTYCKI zrl>+(D*)+n(D)-n(D*) _ However, since J+(Dl) n(Dl) Therefore, only D° < ip+(D) . ^(D)(^°) = (»i)»-W+-W-1. 8 by induction, we need a slightly more general formulation of the lemma. A (connected) arc 7 (with a base point if 7 is closed) in an oriented link diagram D is called a descending part of D if 7 satisfies the following property: If one travels along 7 (according to the orientation of D ) starting from the beginning of 7 (or the base point), then each crossing which is met for the first time is crossed by an overcrossing.

2. 5 that max degv PL(V,Z) +(D). 3 implies that max degv PD(V,Z) = < (j)+(D) — 4 . Therefore, we have ind D = 1. Suppose now that T(D) has no locally maximal subgraphs. 3 that T(D) has a subgraph of type H2 . Then the argument used in the proof of (3) shows that none of the leaves in the resolving binary tree contributes the term o,tf>+(D)-2{z) 5 s i n c e the graphs associated with these leaves are neither of type Hi nor single-edge graphs. Therefore max degv PL(V,Z) < (f)+(D) — 2. 1 3 Let D be a special alternating (positive) diagram of an oriented link L .

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An Index of a Graph With Applications to Knot Theory by Kunio Murasugi


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