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

Delphine Moussard

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 $Δ$-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.

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

Abstract

The -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 -genus. We generalize this to provide a -dimensional characterization of the slice genus. Further, we prove that the -genus majors the -dimensional positive clasp number and we deduce that the difference between the -genus and the slice genus can be arbitrarily large. We introduce the ribbon counterpart of the -genus and prove that it is an upper bound for the ribbon genus. Interpreting the -genera in terms of -distance, we show that the -genus and the ribbon -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 -component link as the algebraic intersection number of three immersed disks bounded by the three components.

Paper Structure

This paper contains 10 sections, 37 theorems, 1 equation, 24 figures.

Table of Contents

  1. Introduction
  2. Definitions and main statements
  3. Surfaces, cobordisms and projections
  4. Case 1:
  5. Case 2:
  6. Case 3:
  7. $T$--genera and $\Delta$--distance
  8. Arf invariant and Milnor's triple linking number
  9. Examples
  10. Colored links

Key Result

Theorem 1

The slice genus of an algebraically split link $L$ equals the minimal genus of a marked normal singular complex for $L$ with no clasp and no borromean triple point.

Figures (24)

  • Figure 1: Lines of double points and their preimages
  • Figure 2: Triple points on a disk The picture represents the singular set of the disk on its preimage. The points $p_1$, $p_2$ and $p_3$ are the three preimages of a triple point $p$.
  • Figure 3: Borromean and non borromean triple points On the left hand side (resp. right hand side), a borromean link (resp. a trivial link) bounds a disks complex with three ribbons and one borromean (resp. non borromean) triple point
  • Figure 4: A branched point
  • Figure 5: Double points lines on a normal singular surface (preimages)
  • ...and 19 more figures

Theorems & Definitions (58)

  • Theorem : Corollary \ref{['corCharSliceGenus']}
  • Theorem : Corollary \ref{['corgenusTgenus']}
  • Theorem : Theorem \ref{['thDeltaDistance']}
  • Corollary 2.1
  • Lemma : Lemma \ref{['lemmagscb']}
  • Theorem : Corollary \ref{['corTgenusclaspnb']}
  • Theorem 2.2: Daemi--Scaduto
  • Corollary 2.3
  • Proposition 3.1
  • proof
  • ...and 48 more