Abstract:
An extension of the Gordon-Litherland pairing to knots in thickened surfaces.The Gordon-Litherland pairing of knots in $S^3$ is a symmetric bilinear form that unifies the quadratic forms of Trotter and Goeritz. More recently, the Gordon-Litherland pairing was extended to knots in $\mathbb{Z}_2$-homology $3$-spheres by J. Greene. Here we consider the case of knots and links in thickened surfaces $\Sigma\times [0,1]$, where $\Sigma$ is closed and oriented. For an unoriented spanning surface $F$ of a $\mathbb{Z}_2$-homologically trivial knot in $\Sigma \times I$, we define a symmetric bilinear form $\mathscr{G}_F:H_1(F) \times H_1(F) \to \mathbb{Z}$. The resulting signature invariants depend only on the $S^*$-equivalence class of $F$. Previously, Im-Lee-Lee defined signature invariants for checkerboard colorable knots in thickened surfaces using a combinatorial Goeritz matrix approach. For $\mathbb{Z}$-homologically trivial knots in $\Sigma \times I$, signature invariants were defined by Boden-Chrisman-Gaudreau via a modification of the Seifert pairing. Our extended Gordon-Litherland pairing $\mathscr{G}_F$ likewise unifies both these signature invariants for knots in $\Sigma \times I$. In particular, for knots in $S^2 \times I$, it agrees with the usual Gordon-Litherland pairing. Furthermore, we show that the extended Gordon-Litherland pairing can be realized as a certain relative intersection form of a twofold branched cover of $F$ pushed into a thickened $3$-manifold $W \times I$, where $\partial W=\Sigma$. We will also discuss some geometric applications, such as computing crosscap numbers of knots in $\Sigma \times I$ and virtual knots. This talk is based on joint work with H.U. Boden and H. Karimi.