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Braid-isotopic closed braids are conjugate

Statement

Assume the Axiom of Choice. Let $β, β^{'}$ be $n$ -braids and suppose $\hat{β}$ and $\hat{β^{'}}$ are isotopic through closed $n$ -braids about the standard axis (an isotopy of the closed braids through braids in the complement of the axis). Then $β^{'}$ is conjugate to $β$ in $B_{n}$ .

Facts & Assumptions

Given: AC, two $n$ -braids $β, β^{'}$ based at $Q$ , and an isotopy of their closures through closed $n$ -braids about the standard axis $A$ .

[F1]

Every closed $n$ -braid about $A$ meets each page in exactly $n$ points, and the closure of a braid based at $Q$ is obtained from its strand images by the fixed diffeomorphism $φ$ of the closure construction (The closure of a geometric braid).

[F2]

The unordered configuration space $C_{n} (D^{\circ})$ of $n$ points in the open disk is the quotient of the ordered configuration space by the label permutations, with basepoint the orbit $[Q]$ of the base configuration; its points are $n$ -element subsets (Unordered configuration spaces $C_{n} (X)$ ).

[F3]

Raw slicing induces a bijection $S$ from the geometric braid group $G_{n}$ to $π_{1} (C_{n} (D^{\circ}), [Q])$ , and the map $Φ ⁣ : G_{n} \to π_{1} (C_{n} (D^{2}), [Q])$ , $[β] \mapsto (ι_{*}^{C} [S (β)])^{- 1}$ , is a group isomorphism (Geometric braid classes and the unordered configuration fundamental group).

[F4]

Assume AC: the published surjection $B_{n}^{A r t i n} \to G_{n}$ is an isomorphism, so the Artin braid group $B_{n}$ of The braid group by Artin presentation is identified with the geometric braid group (The Artin presentation is complete for geometric braids).

[F5]

Two loops in a path-connected space are freely homotopic if and only if their classes in the fundamental group are conjugate; free homotopy classes correspond bijectively to conjugacy classes (Free homotopy classes of loops are conjugacy classes, Based loops and the fundamental group).

[F6]

AC implies countable choice, so the general continuous braid's chosen smooth closure model is available under the Given hypothesis (AC implies DC implies countable choice).

Proof

technique · direct

1.1F1F2F3F4F6

The loop of a closed braid. Fix a page $P_{θ}$ of the closure construction and identify it with the open disk $D^{\circ}$ ; a closed $n$ -braid about $A$ meets each page in exactly $n$ points by [F1], and as the page turns once about the axis these points trace a continuous loop $c ⁣ : S^{1} \to C_{n} (D^{\circ})$ in the unordered configuration space of [F2]. For a general continuous braid use the selected smooth model of [F1], available by [F6]. It is endpoint-fixed braid-isotopic to the given braid, so raw slicing of [F3] sends its geometric class to the class of this closure loop. The group isomorphism in [F3] is instead $Φ (β) = (ι_{*}^{C} [c])^{- 1}$ . Apply that fixed induced map and inversion when transporting conjugacy; both carry conjugate classes to conjugate classes. Composing $Φ$ with [F4] gives the fixed group isomorphism used below.

1.2F1F2given

Isotopy of closed braids gives a homotopy of the loops. An isotopy of $\hat{β}$ to $\hat{β^{'}}$ through closed $n$ -braids about $A$ gives, by reading the $n$ intersection points with the turning page at each stage, a homotopy of the loops $c$ and $c^{'}$ in $C_{n} (D^{\circ})$ : the intersection points depend continuously on the isotopy parameter because the isotopy is continuous and the page meets every intermediate closed braid transversely in exactly $n$ points. Hence $c$ and $c^{'}$ are freely homotopic loops.

2.1F4F5F6step 1.1step 1.2∎

Conclusion. By [F5] the free homotopy of step 1.2 makes the classes of $c$ and $c^{'}$ conjugate in $π_{1} (C_{n} (D^{\circ}), [Q])$ ; first applying $ι_{*}^{C}$ and inversion, and then transporting through the fixed isomorphisms of step 1.1 gives that $[β]$ and $[β^{'}]$ are conjugate in $B_{n}$ , so $β^{'}$ is conjugate to $β$ in $B_{n}$ . The axiom of choice supplies [F4], through Artin-presentation completeness, and the countable choice for the smooth closure model in [F6].

Depends on

Used by

Dependency tree · two levels

55 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Birman and Brendle, Braids: A Survey, Handbook of Knot Theory chapter, author manuscript; section 2.3 and the remark attributed there to Morton, printed pp. 18-19 (standard reference, not scraped)