Celebratio Mathematica — Birman

Interview with Joan Birman

Works connected to William Wyatt Menasco

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J. S. Birman and W. W. Menasco: “Stabilization in the braid groups, I: MTWS,” Geom. Topol. 10 : 1 (2006), pp. 413–540. MR 2224463 Zbl 1128.57003 article

Choose any oriented link type \( \mathscr{X} \) and closed braid representatives \( X_+ \), \( X_- \) of \( \mathscr{X} \), where \( X_- \) has minimal braid index among all closed braid representatives of \( \mathscr{X} \). The main result of this paper is a ‘Markov theorem without stabilization’. It asserts that there is a complexity function and a finite set of ‘templates’ such that (possibly after initial complexity-reducing modifications in the choice of \( X_+ \) and \( X_- \) which replace them with closed braids \( X^{\prime}_+ \), \( X^{\prime}_- \)) there is a sequence of closed braid representatives \[ X^{\prime}_+ = X^1 \to X^2 \to \cdots \to X^r = X^{\prime}_- \] such that each passage \( X^i\to X^{i+1} \) is strictly complexity reducing and non-increasing on braid index. The templates which define the passages \( X^i \to X^{i+1} \) include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index \( m\geq 4 \) a finite set \( \mathscr{T}(m) \) of new ones. The number of templates in \( \mathscr{T}(m) \) is a non-decreasing function of \( m \). We give examples of members of \( \mathscr{T}(m) \), \( m\geq 4 \), but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

J. S. Birman and W. W. Menasco: “Stabilization in the braid groups, II: Transversal simplicity of knots,” Geom. Topol. 10 : 3 (2006), pp. 1425–1452. MR 2255503 Zbl 1130.57005 ArXiv math.GT/0310280 article