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Signature of rotors. (English) Zbl 1079.57007

Suppose that for a link diagram \(D\), there exists an \(n\)-gon \(R\) such that \(R\cap D\) consists of \(n\) arcs and possibly circles, and that \(R\cap D\) is invariant under the \(2\pi/n\)-rotation. A rotant of \(D\) is a reflection of \(R\cap D\) along an axis keeping the outside of \(R\) fixed, see R. P. Anstee, J. H. Przytycki and D. Rolfsen [Topology Appl. 32, No.3, 237–249 (1989; Zbl 0638.57006)]. Note that all reflections give the same link, and that each side of \(R\) intersects \(D\) in two points. A rotant is called orientation-preserving (orientation-reversing, respectively) if the two arcs near these points are antiparallel (parallel, respectively).
In the paper under review the authors prove that the characteristic polynomial of a Goeritz matrix does not alter by taking rotants and that a Hermitian matrix defined by a Seifert matrix does not alter by taking oriantation-preserving rotants. Here a Seifert matrix \(A\) and a complex number \(\xi\) define a Hermitian matrix \(\xi A+ \overline{\xi}A^{t}\), where \(\overline{\xi}\) denotes the complex conjugate and \(A^{t}\) is the transposed matrix of \(A\). As corollaries they show that the Murasugi signature for unoriented links is not changed by taking rotants and that the Tristram-Levine signature for oriented links is not changed by taking orientation-preserving rotants.
Moreover they show that there exists a pair of orientation-reversing rotants with different Conway polynomial. Note that P. Traczyk proved that the Conway polynomial does not alter by taking orientation-preserving rotants [P. Traczyk, Geom. Dedicata 110, 49–61 (2005; Zbl 1081.57007)].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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