Boundary slopes of non-orientable Seifert surfaces for knots
A Seifert surface for a knot \(K\) in the 3-sphere \(S^3\) is a connected compact surface with boundary \(K\). A slope is the isotopy class of an essential unoriented simple curve on the torus \(\partial E(K)\), where \(E(K)= \text{cl} (S^3- N(K))\) and \(N(K)\) is a tubular neighborhood of \(K\). For a Seifert surface \(F\), \(F\cap \partial E(K)\) is an essential loop in \(\partial E(K)\) and it defines a slope, called the boundary slope of \(F\). Usually, a Seifert surface is assumed to be orientable but any knot admits also a non-orientable one. The crosscap number of a knot is the minimal number of the first Betti numbers of non-orientable Seifert surfaces of the knot, and a non-orientable Seifert surface is said to be minimal genus if its first Betti number equals the crosscap number of the knot. In this paper the authors consider minimal genus non-orientable Seifert surfaces and study their boundary slopes. In particular, they prove that for a crosscap number two knot there are at most two slopes which can be the boundary slope of its minimal genus non-orientable Seifert surface and describe an infinite family of knots with two such slopes. Moreover, they discuss the existence of essential non-orientable Seifert surfaces for knots and show that every knot whose crosscap number is at most two has an essential non-orientable Seifert surface.
scientific article; zbMATH DE number 1787579
scientific article