RESEARCH STATEMENT THOMAS KINDRED My approach to math research is highly visual, explicitly constructive, often combinatorial, sometimes com- putational: ask intuitive but challenging questions about deep ideas; use elementary methods, especially constructive, visual, and/or combinatorial ones, to attack these questions; persist; engage fully with the writing and revision process, refining insights to a level of clarity where they seem obvious; use this clarity to look for further insights, perhaps computational; share; ask more questions; continue in this way. My goal is not just to prove, but to understand and convey. My research stems, at least indirectly, from a fascination with spanning surfaces in S3, which occupy a perfect intersectional space, relative to my strengths and interests, from which to reach out to diverse areas: • Spanning surfaces are versatile: nearly every aspect of low-dimensional topology can be understood in terms of knots and links in S3, and spanning surfaces offer a visual, constructive, geometric approach to nearly every part of knot theory. The basic idea is that the space around a knotted circle carries a surprising wealth of mathematical data, and spanning surfaces help parse this data. • Spanning surfaces invite explicitly constructive approaches, often involving elementary methods, often combinatorial in nature. These all play to my relative strengths as a researcher. • Spanning surfaces are fun! You draw them. You can hold them in your hands. Their visual appeal conceals their profound mathematical relevance in a way that reminds me, by analogy, of a quotation: Music is the pleasure the human mind experiences from counting without being aware that it is counting { Gottfried Liebniz Organization: Section 1 surveys the mathematical and historical foundations of my research. I hope that mathematicians from all backgrounds find it accessible. It can be treated as a self-contained survey of the main themes of my research. Section 2 surveys my eight completed papers, and my ninth paper, in preparation, and describes ways that I plan to extend them. Section 3 describes other likely directions for my future research. First, though, I want to highlight a recent result which conveys the overall flavor of my research. As I will describe further in xx1-2, in 1993 Menasco and Thistlethwaite used the Jones polynomial to give the first proof of Tait's 1898 flyping conjecture. I recently gave the first purely geometric proof of their theorem: Flyping Theorem: Any two reduced alternating diagrams of a given prime nonsplit link are related by a sequence of flype moves. My proof stems from the insight that a flype move on a diagram corresponds to a geometric operation, which I call re-plumbing, on one of that diagrams' chessboard surfaces (and an isotopy of the other chessboard surface). Figure 1 illustrates this correspondence. This particular insight and the approach to the flyping theorem that grew out of it (and less directly all of my research) all grew out of my first week conducting undergraduate research, with Colin Adams at Williams' REU SMALL in 2005. 1 2 THOMAS KINDRED T T 2 2 2 T T T 1 1 Figure 1. A flype move (top) on a link diagram corresponds to a re-plumbing move (bot- tom) on one of its chessboard surfaces (here, the black surface) and an isotopy of the other surface (here, the white surface). 1. Mathematical and historical background A mathematical knot is an embedded circle in 3-dimensional space S3; a knot consisting of more than one circle is called a link. Most of what follows works for links, but I will focus on knots, where the definitions and theorems tend to be more straightforward. The first knot theorist, Gauss, described how to rep- resent knots diagrammatically, as in Figure 1. Gauss also described how to represent any knot diagram by a sequence of positive integers: pick a basepoint and orient the knot; then walk along the knot, recording each new crossing with the next unused integer and each repeated crossing with the corresponding inte- 1 ger. To this day, the interplay between the combina- Figure 2. An alternating knot dia- torics of knot diagrams and the topology of knots and gram. Using the indicated basepoint manifolds often reveals new deep and surprising con- and orientation, the Gauss code is 2 nections (e.g. see x2.7), and Gauss codes are perfect (1; 2; 3; 1; 4; 5; 6; 3; 2; 4; 7; 6; 5; 7). for computing (see x2.5 for examples). The first systematic treatment of knot theory came almost a century later, when in 1898 Tait constructed a table of all alternating knots through ten crossings. Tait justified his tabulation entirely on intuition (the hard part was proving that knots that seem to be different actually are different). Tait conjectured [64]: • Any reduced3 alternating diagram realizes its knot's (minimal) crossing number. • All reduced alternating diagrams of a given alternating knot have the same writhe.4 1In general, one uses the over/under information at the crossings to attache signs to these integers; when the diagram alternates between over- and under-crossings, this is unnecessary, subject to the convention that the first crossing is an over-crossing. 2and DT codes, which are half the length but carry the same information 3A knot diagram is reduced if it has no \nugatory" crossing: or more generally . 4Orient a knot diagram D arbitrarily. Then every crossing looks like or . The writhe of D is j j − j j. RESEARCH STATEMENT 3 • All reduced alternating diagrams of any prime5 alternating knot are related by flypes (see Fig. 1). Interestingly, Tait was motivated by an erroneous belief in the vortex theory of atoms, which held that atoms were knotted circles in the ether; Tait hoped that his table might correspond to a table of the elements. Before long, physics abandoned the theory of the ether, in favor of Einstein's general theory of relativity, and the vortex theory of atoms, in favor of quantum mechanics. Still, knot theory soon found new motivation in Dehn's insight (circa 1910) that, by drilling out (thickened) knotted circles and gluing them back in \another way," one could construct many different 3-manifolds. In 1960-62, Lickorish and Wallace used Heegaard splittings and mapping class groups of surfaces to prove that every closed orientable 3-manifold can be obtained via such Dehn surgery, thus implying that the study of 3-manifolds corresponds to the study of integer-weighted links in S3; the integer tells one how to glue the thickened knot back in. The advent of Kirby calculus in 1978 created a similar correspondence between closed oriented smooth 4-manifolds and integer-weighted links, this time in connected sums of S1 ×S2. The connection here is that performing Dehn surgery on S3 = @D4 corresponds to attaching a 2-handle to D4. (More on 4-manifolds at the end of x1.) Despite these fantastic reasons to study knots, Tait's conjectures all remained unproven into the mid-80's. This was especially striking, given that his tabulation was verified using discoveries from the 1920's and 30's: • Reidemeister described three types of moves that relate all diagrams of a given knot. These moves provide a rigorous framework for developing knot invariants: numbers, polynomials, groups, etc. that one assigns to a knot in a way that doesn't depend on the diagram. • Seifert proved algorithmically that every knot K is the boundary of a connected orientable (i.e. 2-sided) surface F embedded in S3. Such F is now called a Seifert surface for K. The minimum genus among all Seifert surfaces for K is called its classical genus, denoted g(K). • Alexander and Briggs discovered the first polynomial invariant for knots. One can compute the Alexander polynomial ∆(K) directly from a knot diagram, but ∆(K) has a geometric interpreta- tion: construct the infinite cyclic cover X of S3 nL by cutting along a Seifert surface, taking infinitely −1 many copies, and gluing them together; then H1(X) is isomorphic to Z[t; t ]=∆(K). In addition to verifying Tait's tabulation, these discoveries opened up, and in some cases solved, new ques- tions. For example, in the 1950's, Crowell and Murasugi independently used ∆(K) to prove that applying Seifert's algorithm to any reduced alternating knot diagram yields a surface that realizes g(K) [18, 55]. Meanwhile, the two theories that had displaced the theory of the ether and atomic vortices had come into conflict, creating what to this day remains the biggest open problem in physics: how to \quantize gravity"? This would unify quantum mechanics and relativity, the theories which respectively govern the cosmically large and the immeasurably small. In the context of this conundrum and Tait's still-unproven conjectures, a pair of revolutionary discoveries further ignited interest in the theory of knots. First, in the late 1970's, Thurston realized that \most" knot complements admit a (unique) hyperbolic structure. Hyperbolic geometry, arises naturally in relativity as a \slice of the future," the set of all points with a given time-like separation from, say, here and now. Second, Jones' work in operator algebras (a vital tool in quantum mechanics) led in 1985 to the discovery of a new polynomial invariant for knots [39].6 The Jones polynomial quickly led to several proofs of Tait's conjectures about crossing number and writhe [65, 42, 57, 68]. The proofs were surprisingly straightforward, 5A knot K is prime if it cannot be written as a connected sum, i.e. if, whenever an embedded sphere Q ⊂ S3 intersects K transversally in two points, one of the two arcs of K n Q is unknotted.