Mathematical Sciences: Polynomial Invariants in the Theory of Knots

点击翻译按钮获取中文摘要

This project concerns polynomial invariants in knot theory, particularly the Jones polynomial and its generalizations. Along with knot theory, the project is integrally involved with related problems in combinatorics, generalized spin networks, quantum groups, statistical mechanics, and invariants of 3-manifolds. In particular, it addresses a tangle-theoretic recoupling theory for the Temperley-Lieb algebra and its relationship with invariants of graph embeddings, the SL(2) quantum group, and the Turaev-Viro invariant of 3-manifolds. This recoupling theory gives a completely tangle-theoretic version of quantum 6j coefficients at roots of unity, and it permits the construction of the Turaev-Viro invariant on a purely knot-theoretic basis. These topics will be explored in the context of the Witten-Reshetikhin-Turaev invariants of 3-manifolds. Other problems about link polynomials will be considered, such as the properties of statistical mechanical models for the Alexander-Conway polynomial and relations with four- dimensional topology. 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. 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. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.