The theme of this project is understanding how certain types of surfaces and 2-dimensional objects can be embedded, immersed, and, in general, positioned in the 3-dimensional sphere. Three ongoing collaborative projects and one individual one are involved, concerning (1) links represented by closed braids, with Joan S. Birman of Columbia University, (2) planar surfaces and the cabling conjecture, with Abigail A. Thompson of the University of California, Davis, (3) embeddings and immersions in highly alternating link exteriors, with Morwen B. Thistlethwaite of the University of Tennessee, and (4) the Milnor-Bennequin conjecture and unknotting number. (1) seeks better ways of calculating about such links, possibly leading to a more efficient effective algorithm than Haken's for distinguishing oriented links. (3) builds upon a recent joint triumph in settling the hundred-year-old Tait flyping conjecture. Knots and their generalization, links, 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. Topologists say that they are two different geometric realizations of the same topological knot. 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. The investigator has been in the thick of this activity and has been particularly successful in applying some new polynomial invariants to settle old questions about alternating knots. He will continue to exploit these techniques as well as some more geometric ones.
