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Operadic actions on long knots and 2-string links

Etienne Batelier, Julien Ducoulombier

Abstract

In the present work, we realize the space of 2-string links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of $\mathcal{L}$ by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a 2-string link in terms of the homotopy types of the classes of its prime factors.

Operadic actions on long knots and 2-string links

Abstract

In the present work, we realize the space of 2-string links as a free algebra over a colored operad denoted (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a 2-string link in terms of the homotopy types of the classes of its prime factors.

Paper Structure

This paper contains 17 sections, 27 theorems, 64 equations, 18 figures.

Table of Contents

  1. Embedding spaces
  2. Knot spaces
  3. Link spaces
  4. Operads
  5. Colored operads and their algebras
  6. Free algebras
  7. The little cubes operad
  8. The operad $\mathcal{SCL}$
  9. Operadic actions
  10. Budney's action on fat long knots
  11. Burke and Koytcheff's actions on fat $2$--string links
  12. The action of $\mathcal{SCL}$ on fat $2$--string links
  13. Freeness results
  14. Notions of $3$--dimensional topology
  15. Budney's freeness theorem
  16. ...and 2 more sections

Key Result

Theorem 1

The family $(\hat{\mathcal{L}},\hat{\mathcal{K}},\hat{\mathcal{K}},\hat{\mathcal{K}})$ is an $\mathcal{SCL}$--algebra. In particular, the family $(\mathcal{L},\mathcal{K},\mathcal{K},\mathcal{K})$ is homotopy equivalent to an explicit $\mathcal{SCL}$--algebra.

Figures (18)

  • Figure 1: Illustration of the connected sum of two knots.
  • Figure 2: Illustration of the commutativity in the monoid $\pi_0 \mathcal{K}$.
  • Figure 3: A framed long knot $f$ with framing number $4$.
  • Figure 4: The two top string links are not isotopic (closing the first one up with two vertical lines results in a trivial knot, doing so with the second one yields a trefoil knot). The corresponding links are however isotopic.
  • Figure 5: Illustration of the commutation between a braid and an arbitrary string link.
  • ...and 13 more figures

Theorems & Definitions (74)

  • Theorem : Theorem \ref{['SCLActs']}
  • Theorem : Theorem \ref{['freeSCL']}
  • Theorem 1.1: Schubert Schubert
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4: Budney Budney
  • Definition 1.5
  • Definition 1.6: Budney Budney
  • Proposition 1.7: Budney Budney
  • Corollary 1.8
  • ...and 64 more