Slice genus, $T$-genus and $4$-dimensional clasp number

TL;DR

This paper introduces the $T$-genus as a new invariant bounding the slice genus of knots, explores its properties, and relates it to ribbon and clasp numbers, providing new characterizations and bounds in 4-dimensional knot theory.

Contribution

It generalizes the $T$-genus to characterize the slice genus and establishes its relation to clasp and ribbon genera, also extending the framework to links and colored links.

Findings

$T$-genus bounds the slice genus and positive clasp number.

$T$-genus and ribbon $T$-genus coincide iff all slice knots are ribbon.

Difference between $T$-genus and slice genus can be arbitrarily large.

Abstract

The $T$-genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their $T$-genus. We generalize this to provide a $3$-dimensional characterization of the slice genus. Further, we prove that the $T$-genus majors the $4$-dimensional positive clasp number and we deduce that the difference between the $T$-genus and the slice genus can be arbitrarily large. We introduce the ribbon counterpart of the $T$-genus and prove that it is an upper bound for the ribbon genus. Interpreting the $T$-genera in terms of $\Delta$-distance, we show that the $T$-genus and the ribbon $T$-genus coincide for all knots if and only if all slice knots are ribbon. We work in the more general setting of algebraically split links and we also discuss the case of colored links. Finally, we express Milnor's triple linking number of an algebraically split $3$-component link as the algebraic intersection number of three immersed disks bounded by the three components.