Narrow Results

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^{\ast }$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^{\ast }$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

We consider Seifert surfaces of knots in a spatial graph whose interiors are mutually disjoint. We give the upper bound of the number of such surfaces. Then we show that for a given abstract graph there is a spatial embedding of the graph which realizes the upper bound.

Gordon and Litherland showed that all compact, unoriented, possibly non-orientable surfaces in S³ bounded by a link are related by attaching/deleting tubes and half twisted bands. In this note we give an elementary proof for this result.

Using a special form of spanning surface for a knot, we give a formula for the coefficient of the $z^2$-term of the Alexander–Conway polynomial in terms of the sum of determinants of the blocks of $2\times 2$ submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

From classical knot theory we know that every knot in $S^3$ is the boundary of an oriented, embedded surface. A standard demonstration of this fact achieved by elementary techniques comes from taking a regular projection of any knot and employing Seifert’s constructive algorithm. In this note we give a natural generalization of Seifert’s algorithm to any closed integral homology 3-sphere. The starting point of our algorithm is presenting the handle structure of a Heegaard splitting of a given integral homology sphere as a planar diagram on the boundary of a $3$-ball. (For a well-known example of such a planar presentation, see the Poincaré homology sphere planar presentation in Knots and Links by Rolfsen 3.) An oriented link can then be represented by the regular projection of an oriented $k$-strand tangle. From there we give a natural way to find a “Seifert circle” and associated half-twisted bands.

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.