Celebratio Mathematica

J. S. Birman: “On Siegel’s modular group,” Math. Ann. 191 (1971), pp. 59–68. MR 0280606 Zbl 0208.10601 article

Siegel’s modular group \( \Gamma_g \) is the group of all \( 2g \times 2g \) matrices with integral entries which satisfy the condition: \[ SJS^{\prime} = J \] where \( s \in \Gamma_g \), \( S^{\prime} = \) transpose of \( S \), and if \( I_g \) and \( 0_g \) are the \( g\times g \) unit and zero matrices respectively, then: \[ J = \left( \begin{array}{cc} 0_g & I_g \\ -I_g & 0_g \end{array} \right). \] Generators for \( \Gamma_g \) were first determined by Hua and Reiner [1949], and Klingen [1961] obtained a characterization of \( \Gamma_g \) for \( g\geq 2 \) by a finite system of defining relations. However, Klingen’s results have been of somewhat limited use because, while he gives a procedure for finding defining relations, he does not carry it through explicitly, and in fact the explicit determination of such a system involves a fairly tedious and lengthy calculation. Finding ourselves in the position of needing explicit information about \( \Gamma_g \), we set ourselves the task of reducing Klingen’s results to more useable form. The primary purpose of the paper is to give the results of this calculation (Theorem 1) and to describe the methodology behind our proof.

J. S. Birman and H. M. Hilden: “On the mapping class groups of closed surfaces as covering spaces,” pp. 81–115 in Advances in the theory of Riemann surfaces (Stony Brook, NY, 1969). Edited by L. V. Ahlfors, L. Bers, H. M. Farkas, R. C. Gunning, I. Kra, and H. E. Rauch. Annals of Mathematics Studies 66. Princeton University Press, 1971. MR 0292082 Zbl 0217.48602 incollection

Irwin Kra	Related
Hershel Mark Farkas	Related
Harry Ernest Rauch	Related
Lipman Bers	Related
Lars Valerian Ahlfors	Related
Robert Clifford Gunning	Related
Hugh Michael Hilden	Related

J. S. Birman: Braids, links, and mapping class groups. Annals of Mathematics Studies 82. Princeton University Press, 1974. Based on lecture notes by James Cannon. An erratum to Theorem 2.7 is given in Can. J. Math. 34:6 (1982). MR 0375281 Zbl 0305.57013 book

J. S. Birman and R. Craggs: “The \( \mu \)-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold,” Trans. Am. Math. Soc. 237 (March 1978), pp. 283–309. MR 0482765 Zbl 0383.57006 article

Let \( \mathcal{H}(n) \) be the group of orientation-preserving self-homeomorphisms of a closed oriented surface \( \operatorname{Bd} U \) of genus \( n \), and let \( \mathcal{H}(n) \) be the subgroup of those elements which induce the identity on \( H_1(\operatorname{Bd} U;\mathbf{Z}) \). To each element \( h \in \mathcal{H}(n) \) we associate a 3-manifold \( M(h) \) which is defined by a Heegard splitting. It is shown that for each \( h\in\mathcal{H}(n) \) there is a representation \( \rho \) of \( \mathcal{H}(n) \) into \( \mathbf{Z}/2\mathbf{Z} \) such that if \( k\in\mathcal{H}(n) \), then the \( \mu \)-invariant \( \mu(M(h)) \) is equal to the \( \mu \)-invariant \( \mu(M(kh)) \) if and only if \( k\in\operatorname{kernel} \rho \). Thus, properties of the 4-maniolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from \( \mathcal{H}(n) \) onto \( \mathbf{Z}/2\mathbf{Z} \) are studied and are shown to constitute a complete conjugacy class of subgroups of \( \mathcal{H}(n) \). The class has nontrivial finite order.

J. S. Birman and X.-S. Lin: “Knot polynomials and Vassiliev’s invariants,” Invent. Math. 111 : 2 (1993), pp. 225–270. MR 1198809 Zbl 0812.57011 article

J. S. Birman and M. D. Hirsch: “A new algorithm for recognizing the unknot,” Geom. Topol. 2 (1998), pp. 175–220. MR 1658024 Zbl 0955.57005 article

The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.

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

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