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Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus

Junghwan Park, Zhongtao Wu, Jingling Yang

Abstract

Hom and Wu introduced the knot concordance invariant $ν^{+}$ for knots in $S^{3}$ and proved that it gives a lower bound for the slice genus. Wu and Yang extended $ν^{+}$ to knots in rational homology $3$-spheres, where it gives a lower bound for the rational slice genus, an analogue of the slice genus for knots in rational homology $3$-spheres. We call a knot $ν^{+}$-sharp if this bound is realized as an equality. An open question asks whether a local knot in a $3$-manifold $Y$, that is, a knot contained in a $3$-ball, can bound a surface of smaller genus in $Y\times I$ than in $S^{3}\times I$. Using the Heegaard Floer invariant $ν^+$, we show that this does not occur for local knots arising from $ν^+$-sharp knots: if $K\subset S^3$ is $ν^+$-sharp and $Y$ is a rational homology $3$-sphere, then the induced local knot in $Y$ has rational slice genus equal to the slice genus of $K$. The proof proceeds by establishing an additivity result for the rational slice genus.

Local Knots, $ν^+$-Sharp Knots, and Rational Slice Genus

Abstract

Hom and Wu introduced the knot concordance invariant for knots in and proved that it gives a lower bound for the slice genus. Wu and Yang extended to knots in rational homology -spheres, where it gives a lower bound for the rational slice genus, an analogue of the slice genus for knots in rational homology -spheres. We call a knot -sharp if this bound is realized as an equality. An open question asks whether a local knot in a -manifold , that is, a knot contained in a -ball, can bound a surface of smaller genus in than in . Using the Heegaard Floer invariant , we show that this does not occur for local knots arising from -sharp knots: if is -sharp and is a rational homology -sphere, then the induced local knot in has rational slice genus equal to the slice genus of . The proof proceeds by establishing an additivity result for the rational slice genus.
Paper Structure (17 sections, 16 theorems, 113 equations, 1 figure)

This paper contains 17 sections, 16 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational Seifert surfaces and slice surfaces
  4. Knot Floer complexes
  5. $V$-invariants
  6. $\nu^+$-invariants
  7. Symmetries in knot Floer homology and the middle relative $\rm Spin^c$ structure
  8. Subadditivity and additivity of $\nu^+$ and $V$-invariants
  9. Subadditivity of $\nu^+$ and $V$-invariants
  10. Additivity of $\nu^+$-invariants
  11. Additivity of the rational slice genus
  12. Examples of $\nu^+$-sharp knots and Floer simple knots
  13. $\nu^+$-sharp knots in $S^3$
  14. Examples with $|\tau(K)|=g_4(K)$
  15. Examples with $|\tau(K)|<\max\{\nu^+(K), \nu^+(-K)\}=g_4(K)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $K$ be a $\nu^+$-sharp knot in $S^3$ and let $Y$ be a rational homology $3$-sphere. If $K\subset Y$ is the induced local knot, then it is $\nu^+$-sharp and $g_{Y \times I}(K)=g_4(K)$.

Figures (1)

  • Figure 1: A knot for which the rational slice genus can be computed using Corollary \ref{['4gad']}.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: WY
  • Theorem 2.2: WY
  • Remark 2.3
  • Lemma 2.4: WY
  • Lemma 2.5: WY
  • Lemma 2.6
  • ...and 11 more