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A satellite formula for real Seiberg-Witten Floer homotopy types

Jin Miyazawa, JungHwan Park, Masaki Taniguchi

TL;DR

This work develops a real Seiberg--Witten Floer theory framework to study knot satellites with odd winding patterns. By constructing real Floer data and proving an excision theorem, it derives a satellite formula that equates $SWF_R(P(K))$ with $SWF_R(K)$ tensored by $SWF_R(P(U))$, and shows key concordance invariants are determined by the zero-framed surgery $S^3_0(K)$. It applies these tools to show infinite order for broad families of satellites of the $E_{2,1}$ knot, strengthens real $10/8$-type inequalities, and introduces degree-type invariants for homology $S^1\times S^3$ manifolds with PSC obstructions. The results highlight the power of real Floer data in constraining knot concordance and 4-manifold topology via zero-framed surgery and expository cobordisms.

Abstract

We establish a satellite formula for the real Seiberg-Witten Floer homotopy types of knots with odd patterns. Using this, we derive several applications to knot concordance theory. The satellite formula follows from a version of the excision theorem for real Floer homotopy types. Additionally, we show that the concordance invariants arising from real Seiberg-Witten theory depend only on the knot's zero-framed surgery.

A satellite formula for real Seiberg-Witten Floer homotopy types

TL;DR

This work develops a real Seiberg--Witten Floer theory framework to study knot satellites with odd winding patterns. By constructing real Floer data and proving an excision theorem, it derives a satellite formula that equates with tensored by , and shows key concordance invariants are determined by the zero-framed surgery . It applies these tools to show infinite order for broad families of satellites of the knot, strengthens real -type inequalities, and introduces degree-type invariants for homology manifolds with PSC obstructions. The results highlight the power of real Floer data in constraining knot concordance and 4-manifold topology via zero-framed surgery and expository cobordisms.

Abstract

We establish a satellite formula for the real Seiberg-Witten Floer homotopy types of knots with odd patterns. Using this, we derive several applications to knot concordance theory. The satellite formula follows from a version of the excision theorem for real Floer homotopy types. Additionally, we show that the concordance invariants arising from real Seiberg-Witten theory depend only on the knot's zero-framed surgery.

Paper Structure

This paper contains 21 sections, 45 theorems, 194 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Real structures on zero-framed surgery trace
  3. Real spin$^c$ structures
  4. Dimension four
  5. Dimension three
  6. Real structures on surgeries
  7. Preliminaries of real Floer homotopy type for knots
  8. Representations
  9. Construction of real Floer homotopy type
  10. ${\rm Pin}^-(2)$-Monopoles for Homology $S^1 \times S^2$
  11. Real spin structures for homology $S^1 \times S^2$
  12. Real Floer homotopy type for homology $S^1 \times S^2$
  13. Cobordism maps between homology $S^1\times S^2$ manifolds
  14. Results on zero-framed surgery
  15. Excision theorem in real Seiberg--Witten theory
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $P$ is a pattern with an odd winding number and $P(U)$ is slice, then any finite self-connected sum of $P(E_{2,1})$ does not bound a normally immersed disk in the four-ball $B^4$ with only negative double points. In particular, the knot $P(E_{2,1})$ has infinite order in $\mathcal{C}$.

Figures (3)

  • Figure 1: The manifold $U$ with eight codimension-one faces.
  • Figure 2: The manifold obtained by gluing $U \times T^2$ with $Y_1 \times [0,1]$, $Y_1' \times [0,1]$, $-Y_2 \times [0,1]$, and $-Y_2'\times [0,1]$.
  • Figure 3: The manifold $-X^0 \circ X^0$ and fiberwise surgeries.

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 78 more