Mathematical Sciences: Knotting in 3-Manifolds

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Three investigators are engaged in this research project in low-dimensional topology. (i) Scharlemann intends to apply the new research tools of "thin position," combinatorics of surface intersections, and sutured manifold theory to several problems in the theory of knots and graphs in 3-space and of Heegaard splittings of 3-manifolds. Of particular interest are methods for determining and characterizing knotting of graphs in 3-manifolds, for applying these tools to the graphs which arise naturally in Heegaard splittings of 3-manifolds, and, in the relatively simple case in which the graphs consist of simple closed curves, understanding how the topology changes under a process called Dehn surgery. (ii) Long plans to examine several interrelated questions in the theory of low-dimensional manifolds, braid groups, and representation varieties. The specifics concern representations of the fundamental groups of 3-manifolds and applications of these to boundary slopes and Property P, as well as the braid groups and their representations. (iii) Cooper intends to study ties of low- dimensional topology to hyperbolic geometry. The connection between incompressible surfaces in knot complements and the Newton polygons of the SL2(C) knot polynomial provides a method of getting at deep information about the topology of knot complements. The hyperbolic gluing equations for a knot complement provide another means to approach the same problem. Hodgson's formula for the hyperbolic volume form provides a closed 1-form on any curve given by a polynomial in the longitude and meridian eigenvalues. The exactness of this volume form on a curve arising from a knot puts strong conditions on which polynomials arise this way. Following these leads, Cooper expects to find mutant knots with distinct SL2(C) polynomials, even without being able to calculate these polynomials explicitly. Knots are rather elementary geometric objects whose really interesting properties are topological. By this we mean that two geometric knots do not differ in an interesting way if one of them can be transformed to look just like the other without cutting or untying it, just by pushing its string about to rearrange the crossings. Nevertheless, it is not a trivial matter to recognize when one complicated geometric knot is topologically different from another, rather than just a different geometric realization. This problem can be addressed by computing certain numbers or polynomials which are called "topological invariants," meaning that they always have the same value for different geometric realizations of the same topological knot. The problem would be reduced to pure algebra if there were one invariant which also always had different values for geometric realizations of different topological knots, but life is not so simple -- no single invariant achieves this ideal, nor even all the known invariants taken together. It is therefore valuable to investigate new invariants, some of the most useful being those inspired in recent years by ideas from quantum physics. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.