On the \(C_n\)-distance and Vassiliev invariants (Q2909499)

This paper concerns \(C_n\)-moves on knots (which are closely related to Vassiliev invariants [\textit{M. N. Gusarov}, St. Petersbg. Math. J. 12, No. 4, 569--604 (2001; Zbl 0981.57006); \textit{K. Habiro}, Geom. Topol. 4, 1--83 (2000; Zbl 0941.57015)]), and the number of \(C_n\)-moves required to take one knot to another. Let \(K_1\) and \(K_2\) be oriented knots. The \(C_n\)-distance between \(K_1\) and \(K_2\), denoted by \(d_{C_n}(K_1,K_2)\), is defined to be the minimum number of \(C_n\)-moves that take \(K_1\) to \(K_2\).NEWLINENEWLINEThe main result of this paper is as follows. Let \(p,q,m,n\in \mathbb{N}\) with \(p>q\geq1\), and let \(K_1\) and \(K_2\) be oriented knots such that \(d_{C_n}(K_1,K_2) =p\). Then there exists an infinite family of oriented knots \(\{J_j \; | \; j\in \mathbb{N}\} \), such that for each \(j\), \(d_{C_n}(K_1,J_j)=q\) and \(d_{C_n}(J_j,K_2)=p-q\). These families are shown to have the additional properties that, when \(i\neq j\), \(J_i\) and \(J_j\) are not distinguished by Vassiliev invariants of degree at most \(m\); and are distinguished by the Conway polynomial if and only if \(n=1\) or \(n=2\).