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Braid ordering and the geometry of closed braid. (English) Zbl 1214.57010

Using the Dehornoy (left) ordering \(<_{D}\), on the Braid Group \(B_n\) the author gives the definition of the Dehornoy floor \([\beta]_{D}\) of an \(n\)-braid \(\beta\) in terms of the Garside fundamental word
\[ \Delta =(\sigma_{1}\sigma_{2}\dots \sigma_{n-1})(\sigma_{1}\sigma_{2}\dots \sigma_{n-2})\dots(\sigma_{1}\sigma_{2})(\sigma_{1}) \]
as follows:
\[ [\beta]_{D} = \min\{m\in Z_{\geqslant 0}\mid \Delta^{-2m-2} <_{D} \beta <_{D} \Delta^{2m+2}\}. \]
(The author notes that Thurston’s floor may also be used for the theorem that follows.)
The first result relates \([\beta]_{D}\) and the (non-zero) genus \(g\) of an essential surface \(F\) in the complement of the closed braid \(\hat{\beta}\): 6.5mm
(1)
If \(F\) is tiled, then \([\beta]_{D} < g+1\).
(2)
If \(F\) is mixed-foliated, then \([\beta]_{D} < 2g\).
(3)
If \([\beta]_{D} \geqslant 2g\), then \(F\) is circular-foliated.
The foliations in question are those of the Birman-Menasco braid foliation theory (arising from intersections of discs spanning the axis of \(\hat{\beta}\) with \(F\)). The author notes that (3) means there is an isotopy of \(F\) so that it does not intersect the axis of \(\hat{\beta}\), when \([\beta]_{D} \geqslant 2g\).
Nielsen-Thurston classify braids according to their actions when considered as members of the mapping class group of the punctured disc. As such, a braid \(\beta\) may be periodic, reducible, or psuedo-Anosov. If \([\beta]_{D} \geqslant 2\), the author shows (1) \(\beta\) is periodic \(\Leftrightarrow\) \(\hat{\beta}\) is a torus knot. (2) \(\beta\) is reducible \(\Leftrightarrow\) \(\hat{\beta}\) is a satellite knot. (3) \(\beta\) is psuedo-Anosov \(\Leftrightarrow\) \(\hat{\beta}\) is a hyperbolic knot. The author notes that the condition \([\beta]_{D} \geqslant 2\) is best possible.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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References:

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