Seifert hypersurfaces of 2-knots and Chern–Simons functional

Abstract

We introduce a real-valued functional on the $SU(2)$-representation space of the knot group for any oriented $2$-knot. We calculate the functionals for ribbon $2$-knots and the twisted spun $2$-knots of torus knots, $2$-bridge knots and Montesinos knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of $K$. As a corollary, we show that every oriented $2$-knot having a homology $3$-sphere of a certain class as its Seifert hypersurface admits an $SU(2)$-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology $3$-sphere into a negative definite $4$-manifold to $SU(2)$-representations of their fundamental groups. For example, we prove that every closed definite $4$-manifold containing $\Sigma(2,3,5,7)$ as a submanifold has an uncountable family of $SU(2)$-representations of its fundamental group. This implies that every $2$-knot having $\Sigma(2,3,5,7)$ as a Seifert hypersurface has an uncountable family of $SU(2)$-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.