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.2010 Mathematics Subject Classification. 57M25, 57M27.

Embedding spheres in knot traces

“…For a knot K, g top 4 (K) denotes the (topological) 4-genus or slice genus, the minimal genus among compact, oriented, locally flat surfaces in D 4 with boundary K. The (topological) Z-slice genus g Z 4 (K) is the minimal genus among such surfaces whose complement has infinite cyclic fundamental group. Computable upper bounds for g Z 4 (K) are discussed in [FL18], and include 2g…”

“…Let L be the 2(g − h) + 1-component 1-shaking of K. With everything set up as above, we now look at the 2(m + g − h)-Seifert matrix M of L given by V ⊕ g−h j=1 0 1 1 0 by (8.10). To establish that L admits a Z-slice surface of genus h, it suffices to find a 2(m − h) × 2(m − h) Alexander trivial subblock of M by [FL18,Theorem 1].…”

“…B T C , where C is a 2g × 2g matrix and A is Alexander trivial, i.e. det(tA − A T ) is some power of t. From this definition, one can see that g alg (K) gives a lower bound on the topological Z-slice genus of K: A corresponds to a subsurface S A of S with boundary ∂S A = J A , where J A has trivial Alexander polynomial, and so surgering S along J A gives a Z-slice surface for K of genus g. See [FL18] for details on g alg . Proposition 8.4.…”

“…With everything set up as above, we now look at the -Seifert matrix of given by by (8.10). To establish that admits a -slice surface of genus , it suffices to find a Alexander trivial subblock of by [FL18, Theorem 1].…”

On Calculating the Slice Genera of 11- and 12-Crossing Knots

“…The algebraic genus g alg (K) is a knot invariant determined by the S-equivalence class of Seifert forms θ of a knot K, giving an upper bound for g top 4 (K). It was recently introduced by Feller and the first author [6]; we refer to that paper for a detailed treatment, and only briefly state a definition and some properties of g alg…”