Conway polynomial and Magnus expansion (Q2892183)

Short-circuit closure (the oriented version of Plat closure) converts braids to links and maps the union of the pure subgroups \(\bigcup_{m>0} P_m\) onto the set of oriented knots. This mapping converts a knot invariant to a (pure) braid invariant.NEWLINENEWLINEThe author studies this in the case of \(P_3\) and the Conway polynomial. As the Magnus expansion (with values in the algebra of horizontal chord diagrams) is a universal finite type invariant of \(P_m\) the author is able to find an explicit map for the Conway polynomial from the algebra of horizontal chord diagrams on 3 strands to integer polynomials in one variable. The complexification of the map is evaluated on the Drinfeld associator and a conjecture made (and verified for a few values) concerning the resulting (complex) coefficients of the resulting polynomial.