Abstract
By the works of Gusarov [2] and Habiro [3], it is known that a local move called the Cnmove is strongly related to Vassiliev invariants of order less than n. The coefficient of the znterm in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and KCnthe set of knots obtained from a knot K by a single Cnmove. Let [Formula: see text] be the set of the Conway polynomials [Formula: see text] for a set of knots [Formula: see text]. Our main result is the following: There exists a pair of knots K1, K2such that ∇K1= ∇K2and [Formula: see text]. In other words, the CnGordian complex is not homogeneous with respect to Conway polynomial.
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