Formal Knot Theory (Mathematical Notes, No. 30) by Louis H. Kauffman

By Louis H. Kauffman

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K be the standard labelling of U, and let A = A(K) be the Alexander matrix. trom A(K) of index regions by s. and that A(k,s) be a matrix obtained deleting one column of index k Let ~l Let F(k,s) and Rn+2 A(l,O) Det A(k,s). 1, F(l,O) = and one column by deleting these two columns. . For s ~ k ~ r, the relation t [aP-k]cp and therefore pot,k,r (**) [ak-r]F(k,S) _ [ak-S]F(k,r). To see this identity bring into the determinant lng a column of index r. F(k,s) by multiply- Use (*) in the form g1 ven above 60 to replace the relult1ng column of index not equal to can see only a k l~ecit1c the column ot index a column ot index r I or r.

1, 1, 2, 3, 5, 8 ••••• f k , t k+ l , The coefficients themselves are binomial Consider the simplest case of this phenomenon. Each uni- verse in the sequence shown in Figure 15 has F = 1 has only one state for adjacent fixed stars. If we give the universe Ud its maximal star placement of since it d, and let Fd = Fd(Ud ) with thil placement, then- Fd = ~dF = ~d' Since there is no cancellation ot terms in ~d when expanded as a polynomial in B and W, ~d actually lists the states tor this star placement.

In DA. A deepest shell in a shell composition is a shell whose middle, top, and bottom have no riders. Such shells exist since there are a finite number of shells in the composition. It is then easy to see that an elaboration of a deepest shell Via the interaction rules will always admit a clockwise move. Remark. This completes the proof. Figure 9 illustrates some elaborations of riderless shells, and the available clockwise moves. 40 Fi~re 9 We are now readr to approach the proct ot the Clock Theorem.

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