Journal of Knot Theory and Its Ramifications

A spun knotted torus in the 4-sphere is formed by rigidly sweeping a knotted curve along a circle. Alternately, as the knotted curve is swept along the circle we could give it a number of full turns (Dehn twists). We show the resulting knotted torus depends only on the knotted curve and whether the number of turns is even or odd. The even and odd turned spun tori have nondiffeomorphic complements. This is generalized in some cases to include twist spun turned torus knots.

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified.

The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form on non-oriented spanning surfaces. We develop their cobordism properties from this point of view. We give a good way to index these invariants. We also define d-signatures and d-nullities for links which are null-homologous modulo d in a rational homology sphere from the point of view of branched covers. We index d-signatures and d-nullities and develop their cobordism properties. Finally we define Arf invariants (when possible) in a general closed 3-manifold using spin structures.

We extend the Jones polynomial for links in S³ to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S¹×S² the skein module is infinitely generated.

The Sato-Levine invariant for 2-component links with linking number zero is generalized to links with even linking numbers using not necessarily orientable surfaces. It is well defined modulo 4 and completely classifies “unoriented pass equivalence classes” of 2-component proper links. It is shown that it can be expressed in terms of the coefficients of the Conway polynomial and some applications to the Arf invariant are given.