On classical upper bounds for slice genera

Abstract

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

30 pages, 5 figures, comments welcome! V3: Minor changes and corrections; added a footnote sketching a proof for the folklore fact that Taylor's invariant is a lower bound for the topological slice genus

References in corpus (5)

• The degree of the Alexander polynomial is an upper bound for the topological slice genus
• On the topological 4-genus of torus knots
• On calculating the slice genera of 11- and 12-crossing knots
• On 2-bridge knots with differing smooth and topological slice genera
• Positive braid knots of maximal topological 4-genus

Cited by in corpus (5)

• A note on the topological slice genus of satellite knots
• Embedding spheres in knot traces
• On the genus defect of positive braid knots
• 3-braid knots with maximal 4-genus
• Unknotting via null-homologous twists and multi-twists