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

By Kunio Murasugi

This booklet provides a notable program of graph idea to knot idea. In knot conception, there are various simply outlined geometric invariants which are tremendous tricky to compute; the braid index of a knot or hyperlink is one instance. The authors evaluation the braid index for plenty of knots and hyperlinks utilizing the generalized Jones polynomial and the index of a graph, a brand new invariant brought the following. This invariant, that is made up our minds algorithmically, could be of specific curiosity to machine scientists.

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Go is a spanning subgraph of G, and furthermore, Go is strongly excessive. Consider the dual graph GJ of Go- Since G is reducible, there is a vertex v such that G* — v is a tree. But from our construction of Go, we see easily that GJ — v is also a tree, say TQ. Take a stump v* in T 0 . Let D{ be the domain corresponding to v*. Then all but one edge of dD{ are free in Go and hence ' '' — 1 free edges of dD{ belong to S. Note that any domain in R2 — Go has more than 2 sides, and hence ' is a free edge e' on DD{ which satisfies our requirements.

E± must have a common end with either 20 KUNIO MURASUGI AND J O Z E F H. PRZYTYCKI t\ or e2, as was proved before. If the common end is w o r ^ , then it is easy to see that another end of e4 must be joined to v2 or w by an edge, and hence all four singular edges ei, e2, e3 and e4 occur on a subgraph of type Hi. Suppose that the common end of e4 and ei or e2 is either vi or v 4 . If ei, e 2 and e3 occur on a 4-cycle (Fig. 5 (a)), then four edges, ei,e2,e3 and e4 must occur on a subgraph of type Hi.

While min — max f(y,z) Then max - 7 — 3z v. §8 I m p r o v e m e n t of M o r t o n - F r a n k - W i l l i a m s inequalities Let D be an oriented link diagram of L . Let s(D) be the number of Seifert circles in D . We begin with the following well-known theorem. 1 [FW, Mo 2] For any link diagram D of a link L , h(D) - s(D) - f l < min degv PL(v, z) < max degv PL(v, z) < h(D) + s(D) - 1. 2) Equalities in either side hold for some links, but for many links, inequalities are sharp. In this section we will prove a considerable improvement of these inequalities which, combined with Yamada's Theorem [Y], enables us to determine the braid index of many links.

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