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We show that the coe±cients of a reformulation of a Taylor series expansion of the Hoste and Pryzyticki polynomial are Vassiliev invariants. We also show that many other reformulations of the Taylor series expansion have coe±cients that are Vassiliev invariants. We charecterize the ¯rst two coe±cients b1 0(t) and bL 1 (t) for one of these expan- sions. Moreover, the second coe±cient bL 1 (t); which is a type-one Vassiliev invariant, is given two explicit computational formulas, which are easy to calculate. bL 1 (t) is used to give a lower bound for the crossing number of a knot of zero winding number in the solid torus.

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