By Kunio Murasugi

This publication provides a outstanding software of graph idea to knot idea. In knot concept, there are many simply outlined geometric invariants which are tremendous tough 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, is perhaps of specific curiosity to laptop scientists.

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Define b(D) — rnin by(D) , where the minimum is taken over all descending parts 7 of D which contain all negative crossings. 11 will be given by induction on a pair (n(£>), b(D)) , ordered lexicographically. If n(D) = 0 , then PD(v,z) = (lC^f3Lj and hence a^{D)(z) = (-z)~s+1 . 11 holds for this case. 11 holds for any link diagram D' with (n(D'), b(D')) < (n(D), b(D)). e. equal to b(D)) . There are three cases to be considered. C a s e 1 7 contains self-crossings. AN INDEX OF A GRAPH 43 Let p be the first self-crossing of 7 .

C n i . See Fig. 1 AN INDEX O F A GRAPH 39 D DC0« D* nc,c2 Cn'X Dc,c2 • • • D c,c 2 Dc,c2 /Cn1 /CnX Dco,:::con'-D0 Fig. 1 where, for example, DQ 1 ^ 2 ^ 8 denotes the diagram obtained from D by smoothing c\ and changing crossings at C2, and c 3 . e. diagram) obtained from D by smoothing only. Let Dl denote any leaf different from D° . Note that D° and Dl are positive. D), = T/>+(D°). + n(D) - n+(D) = j>+{D). Z}° contributes a maximal term ^ + (D)(^) • On the other hand, the maximal term in CL^+{D){Z) 2:^+(D) in Dl can contribute is 40 KUNIO MURASUGI AND J O Z E F H.

3) satisfies the following formula (1) -PD+(v,z)-vPD_(v,z) (2) If We denote Pj}(y,z) = V L is a trivial knot thenP£,(v,z) = 1. for PL(V,Z) been used to evaluate PL(V,Z) zPDo(v,z) , if necessary, to emphasize that a diagram D has . 3)(1) the skein relation in this paper. PL(V, Z) is an integer Laurent polynomial on two variables v and z . e. /(f, z) G Z[u, v~1, z, z - 1 ] . We write f(v,z) b = ^2 (t)i(z)vl 7 where a{z) ^ 0 ^ b(z), « < b and i(^) G Z [ ^ , ^ - 1 ] . 4) b = max degv f(v,z) a = mindegv f(v,z) b — a = v — span / ( u , 2) Similarly, we can define maxdegz / ( u , z ) , mindegzf(vyz) and 0 — span f(v,z) .

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