By Kunio Murasugi

This ebook offers a awesome software of graph conception to knot conception. In knot idea, there are many simply outlined geometric invariants which are super tough to compute; the braid index of a knot or hyperlink is one instance. The authors evaluation 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 decided algorithmically, could be of specific curiosity to desktop scientists.

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9 (1) It may not exist a diagram D' such that s(D') — s(D) — (ind+(D) ind-(D)) . 6 + ind-T(D). 7) (1). Write D = D1*D2*- • -*£>m as a *-product of D{ . Then T(D) = r ( D i ) * - • - * r ( D m ) . Since T(Di) is a plane bipartite graph, we have ind+(D) Now consider T(Di). ind+Di If ind+(D\) m — J^ ind+(Di). i= i = 0 , we have nothing to do on D\ . Suppose = k > 0 . Then there exists a singular positive edge e and a vertex v , one of two ends of e , such that ind+ (T(Di)/star v) — k — 1 . e corresponds to a crossing c of D\ .

3) (1) yields H+(DX)(Z) ~ a4>+{D)(z) = 2g) + 1 and s(£>+) = s(D) = s(D%) and J+(D%) > J + ( D ) = J+(DC0), we see that $+(D%) < i/>+(D) and *p+(DcQ) + 1 = i/>+(D). 9) and the induction hypothesis for D+ and DQ yield max degz « ^ + ( D ) ( - ) ^ MD)Proof. 6)(i ) If n-(D) = 0 , then D is a collection of disjoint special alternating diagrams. 1 in [Mu 4]. ) We now proceed by induction on n _ ( D ) . Suppose n* = 7i-(D) ^ 0 . Let ci, C 2 , .

Ii) Let a+(D) = max degza+(D)_2(z) . Then c where Emax(D) ftmax(D) edges. +(D)-2,

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