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On symmetries of peculiar modules; or, $δ$-graded link Floer homology is mutation invariant

Claudius Zibrowius

Abstract

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mutation preserves the hat flavour of the relatively $δ$-graded Heegaard Floer theory of links.

On symmetries of peculiar modules; or, $δ$-graded link Floer homology is mutation invariant

Abstract

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mutation preserves the hat flavour of the relatively -graded Heegaard Floer theory of links.

Paper Structure

This paper contains 11 sections, 25 theorems, 106 equations, 38 figures.

Table of Contents

  1. Introduction
  2. Review: peculiar modules
  3. Definition of peculiar modules
  4. Classification results for peculiar modules
  5. Quotient functors for peculiar modules
  6. Generalized peculiar modules
  7. The action of the mapping class group
  8. Linearity
  9. Stabilization
  10. The action of the half-identity bimodule
  11. Conjugation

Key Result

Theorem 1

Let $L$ be a link in a $\mathbb{Z}$-homology 3-sphere. Suppose $L'$ is obtained from $L$ by Conway mutation. Then $\mathop{\mathrm{\widehat{HFL}}}\nolimits(L)$ and $\mathop{\mathrm{\widehat{HFL}}}\nolimits(L')$ agree as relatively $\delta$-graded invariants.

Figures (38)

  • Figure 1: The Kinoshita-Terasaka knot (left) and its Conway mutant (right). These two knot diagrams agree outside the grey disc bounded by the dotted circle; the tangle diagrams within this disc agree up to a rotation by $\pi$.
  • Figure 2: The $(2,-3)$-pretzel tangle (a) and its immersed curve invariant (b). The latter consists of three components, an embedded one (dashed) and two immersed components, each of which carries a (trivial) 1-dimensional local system.
  • Figure 3: The covering space $\eta\colon\mathbb{R}^2\smallsetminus \mathbb{Z}^2\longrightarrow S^2\smallsetminus\textnormal{4}$
  • Figure 4: The lifts of the curves $\mathfrak{i}_n(\frac{0}{1};4,1)$ (top), $\mathfrak{r}(\frac{0}{1})$ (middle) and $\mathfrak{i}_n(\frac{0}{1};2,3)$ (bottom), illustrating Definition \ref{['def:intro:curves']}.
  • Figure 5: Conway mutation. The relabelling of the tangle ends is illustrated here in terms of rotations of the tangle.
  • ...and 33 more figures

Theorems & Definitions (69)

  • Theorem 1
  • Theorem 2: \ref{['thm:MCGaction']}
  • Definition 3
  • Example 4
  • Theorem 5: \ref{['thm:linearCurves']}, \ref{['thm:rigid_curves']}
  • Theorem 6: \ref{['thm:stabilization']}
  • Theorem 7: \ref{['thm:HalfIdBimodAction']}
  • Definition 8
  • Theorem 9: Bigraded conjugation symmetry for horizontal components; \ref{['thm:Conjugation:Horizontals']}
  • Theorem 10: $\delta$-graded conjugation symmetry
  • ...and 59 more