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

April 3, 2017 | Science Mathematics | By admin | 0 Comments

By Kunio Murasugi

This publication offers a outstanding 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 review 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 the following. This invariant, that is decided algorithmically, could be of specific curiosity to machine scientists.

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E. rotation keeping end points a,a',b,b' fixed) will be called the Tait flype. >—l b Fig. 1 The Tait flype preserves the isotopy class of a link, the property "being alternating", the Tait number w(D) of D and the number of Seifert circles of D . By applying Tait nypes if necessary, we can transform a special link diagram D into a nice special link diagram D' . A special diagram is said to be nice if a disk in R2 bounded by a 2-cycle c = {vo, e i , v i , e2, vo} in T(D) has only edges (of the same sign) connecting two vertices vo and vi .

In this new diagram D[ , two Seifert circles represented by v and vo are amalgamated to one circle and hence s ( D j) = s ( D i ) — 1 • Now we see that T^D^) is the one-point union of T(Di)/star v and some multiple- AN INDEX OF A GRAPH 33 edge graph K , where K contains star v — e as a subgraph and m

Dm [Mu 1]. Therefore, the Seifert graph T(D) is written as T(D) = Ti * • • • * T m , where T; is the Seifert graph of Di . Since Di is a link diagram, Ti is bipartite. If each Di is either a positive or negative diagram, then D is called a homogeneous diagram [C]. If a link admits a homogeneous diagram, it is called a homogeneous link. An alternating diagram is homogeneous, but not conversely. Now suppose D = D\ * D2 . 4 implies the following proposition. 2 Let D be a link diagram and D = Di * D2 .

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