Download e-book for kindle: An Index of a Graph With Applications to Knot Theory by Kunio Murasugi

By Kunio Murasugi

This e-book provides a extraordinary program of graph idea to knot conception. In knot thought, 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 evaluation 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 decided algorithmically, is perhaps of specific curiosity to computing device scientists.

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

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6) (i). 8. Proof. 4) (i) We proceed by induction on n _ ( D ) . 8 itself. Now let c be a negative crossing in D . Denote by D+ the diagrams obtained from D by changing the crossing at c , and DQ the diagram obtained from D by smoothing the crossing c. 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.

4) Note that maxdegv (i) maxdegz a^^D){z) (ii) maxdegz a_(D)(z) < PL(V,Z) 37 L, < */>+(D) < +(D) and mindegv ip-(D). PL(V,Z) > -(D) . 4) for alternating links with index 0. 5 Suppose that D is an alternating link diagram. (D)-l c^ + ( D W + ( D ) = |^ J(-l)n- *+-w*-iD) = y0j h(L) = s(D) iff lfind+(D)=0 lther+\J ifind-(D) itherw\j =0 ind D = 0. 4) (i) holds trivially. Similarly, if ind_D a Then —0 ^ 0 , then (ii) holds trivially, since 4>_(D)(z) — 0 • Therefore, we may assume henceforth that ind+D = 0 in a proof of (i).

N . that the collection of these edges S = {e^i . . , e £n,i • • •, n,\n} 1S a We claim maximal set of independent singular edges in G. However, first we claim that S is cyclically independent. 7 a(G) = E H i U l ^ i ~ ! } • P r o o f Denote p = E " = 1 { | | 5 Z ) j | — 1}. First we show that a(G) < p. Let SQ be a maximal set of cyclically independent singular edges in G. Then \SQ\ = a(G). |<9Z}j| — 1 singular edges on dD{ for each i = 1 , 2 , . . , n, and hence, a possible maximal number of singular edge in SQ is p.

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