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.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces S_g × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ³. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

We introduce a polynomial for plane graphs, which is proposed to equal the Dubrovnik polynomial of the corresponding alternating link diagrams via the medial construction. Then using this polynomial we define another polynomial for doubly edge-weighted plane graphs, which has a natural connection with Dubrovnik polynomial of links formed from plane graphs by edge-tangle replacements. In the final, via this connection we give an explanation for a result due to Lipson and compute the Dubrovnik polynomials of classical pretzel links.

We discuss the connection between colorings of a link diagram and the Goeritz matrix.

If $A$ is an abelian group and $\varphi$ is an integer, let $A(\varphi )$ be the subgroup of $A$ consisting of elements $a\in A$ such that $\varphi \cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0={\varphi }_0,{\varphi }_1,\dots ,{\varphi }_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group ${{𝒟}}_A(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A({\varphi }_0),A({\varphi }_1),\dots ,A({\varphi }_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,{\varphi }_1)}\#\cdots \#T_{(2,{\varphi }_{n-1})}$.

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.

In this paper, we introduce the Goeritz matrix for a pseudo-link whose entries lie in the Laurent polynomial ring $\mathbb{Z}[u^{-1},u]$, which generalizes the Goeritz matrix for a classical link. We show that the determinant of a modified Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer coefficients. We also introduce the notions of the signature, determinant, and nullity of pseudo-links. Further, we discuss some properties of the invariants and compute the polynomials and those numerical invariants for several pseudo-knot families.

Im et al. [Signature, nullity and determinant of checkerboard colorable virtual links, J. Knot Theory Ramifications 19(8) (2010) 1093–1114] introduced how to define Goeritz matrices for checkerboard colorable virtual links. In this paper, we extend this for the Goeritz matrices of virtual knots. And we consider its signature and determinant and show they are invariants for virtual knots.