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Homology concordance and knot Floer homology

Irving Dai, Jennifer Hom, Matthew Stoffregen, Linh Truong

Abstract

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology concordance homomorphisms which vanish for knots coming from $S^3$. This shows that the homology concordance group modulo knots coming from $S^3$ contains an infinite-rank summand. The techniques used here generalize the classification program established in previous papers regarding the local equivalence group of knot Floer complexes over $\mathbb{F}[U, V]/(UV)$. Our results extend this approach to complexes defined over a broader class of rings.

Homology concordance and knot Floer homology

Abstract

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of -valued, linearly independent homology concordance homomorphisms which vanish for knots coming from . This shows that the homology concordance group modulo knots coming from contains an infinite-rank summand. The techniques used here generalize the classification program established in previous papers regarding the local equivalence group of knot Floer complexes over . Our results extend this approach to complexes defined over a broader class of rings.

Paper Structure

This paper contains 27 sections, 60 theorems, 274 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological applications
  3. Background on knot Floer homology
  4. Overview and preliminary notions
  5. Preliminary discussion of $\mathbb{X}$-complexes
  6. Homology spheres other than $S^3$: an example
  7. Grid rings and their properties
  8. Graded valuation rings
  9. Grid rings
  10. Complexes over grid rings
  11. The local equivalence group of knotlike complexes
  12. A total order on $\mathfrak{K}$
  13. Standard complexes and their properties
  14. Standard complexes
  15. Ordering standard complexes
  16. ...and 12 more sections

Key Result

Theorem 1.1

For each $(i,j) \in (\mathbb{Z}\times \mathbb{Z}^{\geq 0}) - (\mathbb{Z}^{\leq 0} \times \{0\})$, there is a homomorphism For classes in ${\mathcal{C}_{\mathbb{Z}}} \subset {\widehat{\mathcal{C}}_\mathbb{Z}}$, all homomorphisms of the form $\varphi_{i,j}$ with $j \neq 0$ vanish. Hence these descend to homomorphisms Moreover, for knots in $S^3$, the remaining homomorphisms $\varphi_{i,0}$ agree w

Figures (3)

  • Figure 1: The homogenous generators (over $\mathbb{F}$) of $\mathcal{R}_U$ (left) and $\mathcal{R}_V$ (right), displayed in the $(\operatorname{gr}_1, \operatorname{gr}_2)$-plane. Red arrows are drawn as a visual aid to represent the action of $U_B$ (left) and $V_T$ (right), but may also be interpreted as helping describe the total order on the set of homogenous elements. Note (for example) that in $\mathcal{R}_U$, the first row dominates the second row; that is, any ${W_{B, i}}$ is divisible by any ${U_B^j}$.
  • Figure 2: The total order $\leq^!$ on $\mathbb{Z} \times \mathbb{Z} - \{(0,0)\}$. Following arrows in the diagram corresponds to decreasing in the total order. Note that $(1,0)$ is the greatest element and $(-1,0)$ is the least.
  • Figure 3: Left: total order on $\Gamma(\mathbb{F}[U])$, with largest element $U$. Successively smaller elements are obtained by following the red arrows; the dashed red arrow schematically represents the fact that $\Gamma_+(\mathbb{F}[U])$ dominates $\Gamma_-(\mathbb{F}[U])$. Although $1$ is not drawn within the total ordering, we may place it in the middle of the dashed arrow. Right: total order on $\Gamma(\mathcal{R}_U)$, with largest element $U_B$. Again, $1$ may be placed in the middle of the long dashed arrow running from the bottom to the top of the diagram. For brevity, we have only labeled elements along the primary axes.

Theorems & Definitions (183)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.3
  • Corollary 1.4
  • Proposition 1.4
  • Corollary 1.5
  • proof
  • Proposition 1.6
  • Remark 1.7
  • ...and 173 more