There are a number of specialties in low-dimensional topology that can find in their "family tree" a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. But its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology. Although the scene of this area has been changed dramatically and experienced significant expansion since the original publication of Professor Joan Birman's seminal work, Braids, Links, and Mapping Class Groups (Princeton University Press), she brought together mathematicians whose research span many specialties, all of common lineage.

The topics covered are quite diverse. Yet they reflect well the aim and spirit of the conference in low-dimensional topology held in honor of Joan S.Birman's 70th birthday at Columbia University (New York, NY), which was to explore how these various specialties in low-dimensional topology have diverged in the past 20-25 years, as well as to explore common threads and potential future directions of development.

Titles in this series are co-published with International Press, Cambridge, MA.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

• J. Cantarella, D. DeTurck, and H. Gluck -- Upper bounds for the writhing of knots and the helicity of vector fields
• O. T. Dasbach and B. S. Mangum -- The automorphism group of a free group is not subgroup separable
• R. Ghrist -- Configuration spaces and braid groups on graphs in robotics
• J. Gilman -- Alternate discreteness tests
• S. P. Humphries -- Intersection-number operators for curves on discs and Chebyshev polynomials
• O. Kharlampovich and A. Myasnikov -- Implicit function theorem over free groups and genus problem
• M. E. Kidwell and T. B. Stanford -- On the $z$-degree of the Kauffman polynomial of a tangle decomposition
• W. Li -- Knot invariants from counting periodic points
• X.-S. Lin and Z. Wang -- Random walk on knot diagrams, colored Jones polynomial and Ihara-Selberg zeta function
• F. Luo -- Some applications of a multiplicative structure on simple loops in surfaces
• W. W. Menasco -- Closed braids and Heegaard splittings
• J. H. Przytycki -- Homotopy and q-homotopy skein modules of 3-manifolds: An example in algebra Situs
• T. Stanford and R. Trapp -- On knot invariants which are not of finite type