By Kunio Murasugi

This booklet provides a amazing program of graph thought to knot concept. 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 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 computing device scientists.

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Furthermore, assume that G is non-separable. Then, if G has index 2, G is one of the following graphs: H^^H^ or H5 (depicted in Fig. 8) or a 22 KUNIO MURASUGI AND JOZEF H. PRZYTYCKI properly chosen subgraph of H4 or H5. Fig. 5 is easy but tedious, we omit the details. §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.

Ejt from S. We need the following easy lemma. 24 KUNIO MURASUGI AND J O Z E F H. , has no cut-vertices. DJz be the bounded domains such that Then i t \0( (J Djm)\ = £ (6-9) m=l \dDjm\-2(£- z (J Djm is connected and m=l 1). 9). Details will be omitted. • Now let C be a simple cycle of G of the smallest length on which all edges e i , . . , e*. occur. , D^L, where m ^ 0 for i — 1 , 2 , . . , ^ . 8, we see that i l m=l m=l \C\ = \0( (J D,m )| = J2 \dD^ I - 2(* - 1). Since e^, j = 1 , 2 , . . Mm| - 2} < \C\.

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.