\(SC_n\)-moves and the \((n+1)\)-st coefficients of the Conway polynomials of links
A \(C_{n}\)-move is a combinatorial simplification involving \(n+1\) clasped arcs of a link diagram. For knot diagrams, the \(C_{n}\)-moves generate Vassiliev \((n-1)\)-equivalence, [cf. \textit{Garoufalidis}, ''Knotted graphs and a geometrical technique of \(n\)-equivalence,'' preprint (1995)] and \textit{K. Habiro} [Geom. Topol. 4, 1--83 (2000; Zbl 0941.57015)]. In general, the \(C_{n}\)-moves preserve Vassiliev invariants of order \(<n\) but they do not suffice to generate Vassiliev \((n-1)\)-equivalence, cf. \textit{Y. Ohyama} [J. Knot Theory Ramifications 11, No.~4, 507--514 (2002; Zbl 1002.57027)]. In an earlier paper [Kobe J. Math. 17, No.~1, 99--117 (2000; Zbl 0966.57015)], the author observed that the mod-2 congruence class of the \(n\)-th coefficient of the Conway polynomial, which is an order-\(n\) Vassiliev invariant, is invariant under a \(C_{n}\)-move. In the current paper the author improves this result in a special case: the mod-2 congruence class of the \((n+1)\)-st coefficient of the Conway polynomial is invariant under an \(SC_{n}\)-move, i.e., a \(C_{n}\)-move in which some link component appears precisely once.
scientific article; zbMATH DE number 5734870
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