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 …

We give a new proof that the Levine–Tristram signatures of a link give lower bounds for the
minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded …

We present evidence supporting the conjecture that, in the topological category, the slice
genus of a satellite knot P (K) is bounded above by the sum of the slice genera of K and P …

We prove that the signature bound for the topological 4‐genus of 3‐strand torus knots is
sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4 …

We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal
genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic …

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal
signature invariant. As an application, we determine all positive braid knots with maximal …

This article contains the results of efforts to determine the values of the smooth and the
topological slice genus of 11-and 12-crossing knots. Upper bounds for these genera were …

Let X be a closed indefinite 4-manifold with b+(X)= 3 (mod 4) and with non-vanishing mod 2
Seiberg–Witten invariants. We prove a new lower bound on the genus of a properly …

We study the four-genus of linear combinations of torus knots: g4 (aT (p, q)#-bT (p′, q′)).
Fixing positive p, q, p′, and q′, our focus is on the behavior of the four-genus as a function …

While the problem of computing the genus of a knot is now fairly well understood, no
algorithm is known for its four-dimensional variants, both in the smooth and in the …