By Kunio Murasugi

This booklet offers a impressive program of graph thought to knot idea. In knot idea, there are many simply outlined geometric invariants which are super tricky to compute; the braid index of a knot or hyperlink is one instance. The authors overview 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's made up our minds algorithmically, might be of specific curiosity to laptop scientists.

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**Extra resources for An Index of a Graph With Applications to Knot Theory**

**Example text**

M. D) — X) ind+(Di). what we sought. 7) (3). 6. 10 The series of diagrams (a) — (d) in Fig. 6 for some link. A diagram D has index 2 and s(D) = 6 , but s(D) = s(D") = 4. Note that D' and D' are the same diagram. QZD r(D) 34 KUNIO MURASUGI AND JOZEF H. PRZYTYCKI HD") Fig. 3. 1 max degv P L (u, z) < h(D') + s(D') - 1 = n{D) + s(D) - 1 - 2 ind+T(D), mindegv PL{v,z) > h(D") - s(D") + 1 = n(D) - s(D) + 1 + 2 ind-T(D) and . 3. 4). Then if ind D — ind+D + , we have b(L) = s(D) - ind D. e. s(D) — ind D < h(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. 5) v - span PL(v, z) < 2{s(D) - 1 - ind+T(D) - ind^T(D)}.

6 I n d e x of a reducible g r a p h In the final section of Chapter I, we will determine the index of a particular type of graphs, called reducible. This is one of a few classes of graphs for which their indices are described in a precise formula. 1 A connected plane graph G is called reducible if G has the following property. Let {-Do, -Di, • • •, Dn} be the set of domains in which R2 is divided by G, where Do is the unbounded domain. Then D 1 ? . , Dn can be renumbered, if necessary , in such a way that for i = 1 , 2 , .