SUBJECT AND ACHIEVEMENT OF PAST RESEARCH My research covers several areas of knot theory, with relations to combinatorics, number theory and algebra, which are outlined below (and by no means unrelated). Vassiliev invariants My first work was to improve the upper bound on the dimension of Vassiliev invariants of degree D [St]. The best known previous upper bound was (D-2)!/2 due to Ng. It was known that Vassiliev invariants can be understood combinatorially in terms of chord diagrams modulo the 4T relation. I introduced a certain type of chord diagrams and showed that they generate modulo 4T relations the space of Vassiliev invariants. Then I estimated from above the number of such chord diagrams to D!/1.1^D. Later, Zagier showed that the generating series of the numbers of such chord diagrams occurs in a strange identity related to the Dedekind eta-function. He found the exact asymptotical behaviour of these numbers, improving the number 1.1 to pi^2/6, thus establishing the currently best upper bound. My later work were constructions of knots with Vassiliev invariants of bounded degree and specific properties, like given unknotting numbers, signatures and 4-genera [St12]. I showed the non-existence of Vassiliev invariants that depend on any finite number of link polynomial coefficients (except the Conway/Alexander polynomial). Legendrian knots Legendrian knots are called knots embedded in the standard contact space. There are inequalities relating the Thurston-Bennequin invariant and Maslov number of Legendrian knots and the degrees of the polynomial invariants of the underlying topological knots. Using these inequalities I gave estimates of the invariants of Legendrian negative knots [St9]. This result can be considered as a generalization of Kanda's determination of the maximal Thurston- Bennequin invariant of the negative trefoil. Gauss diagram formulas Fiedler and Polyak-Viro introduced a new approach to defining Vassiliev invariants by explicit formulas. Such formulas involve sums over specific tuples of crossings of a knot or link diagram of functions involving writhes of the crossings (similarly to linking numbers). These formulas proved useful in the study of positive knots (knots with diagrams all of whose crossings are positive). Positive knots and links have been studied, beside because of their intrinsic knot-theoretical interest, with different motivations and in a variety of contexts, including singularity theory, algebraic curves, dynamical systems, and (in some vague and yet-to-be understood way) in 4-dimensional QFTs. Using the Fiedler-Polyak-Viro formulas, I found several inequalities between Vassiliev invariants of positive knots [St6], allowing to exclude certain knots from having this property. Later I sought generalizations of some criteria to almost positive knots [St9]. Canonical Seifert surfaces The set of knot diagrams whose canonical Seifert surfaces (that is, surfaces obtained by Seifert's algorithm) of given (canonical) genus admits a structure of generating series. It allows to prove, for example, that the number of alternating knots of fixed genus grows polynomially in the crossing number [St10,STV]. Non-trivial Jones polynomial problem In 1985 Jones discovered the famous polynomial invariant named after him and asked if it distinguishes all non-trivial knots from the trivial one. His question remains unanswered despite the recent solution for links. I showed that semiadequate links, as defined by Lickorish-Thistlethwaite, have non-trivial Jones polynomial. Montesinos links are semiadequate, and then I showed that so are 3-braid links, so the non-triviality result applies to these classes. Closed 3-braids I classified among closed 3-braid links the braid positive, strongly quasipositive and fibered ones. Then I showed that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. In recent joint work with M. Hirasawa and M. Ishiwata we showed that 3-braid links have a unique incompressible Seifert surface. Knot tables For some time I have been interested in using knot tables, compiled by Hoste, Thistlethwaite, and Weeks, to seek knots with interesting properties, and so to provide examples and counterexamples to problems that have remained inaccessible using (entirely) manual reasoning [St14,St16,St17,St18]. Other topics I have also done some work on unknotting numbers, link polynomials [St13], number theoretic properties of knot invariants, and enumeration problems of links [St15,St19]. PRESENT AND FUTURE RESEARCH I will list now some main topics of research I'm currently interested in, and plan to work on in the visible future. My choice of problems is not fixed, and will also depend on my interaction with other mathematicians I expect to meet. After every topic I will also briefly explain the expected results of its investigation. Hyperbolic volume There have been so far several situations, in which the hyperbolic volume exhibits a relation to a combinatorily defined knot invariant. The most important one is Kashaev's conjecture on values of colored Jones polynomials, popularized by H. Murakami. Another correspondence was observed by Brittenham, namely that the volume is bounded on alternating knots of given genus. His bound can be improved by inequalities of Lackenby-Agol-Thurston involving an invariant of knot diagrams called twist number. These bounds are also related to conjectures of Dunfield, namely that the volume linearly approximates a logarithm of the determinant of alternating knots. I'm currently interested in obtaining and improving such inequalities. Trivalent graphs and enumeration of knots by genus There is also a relation between the Brittenham approach and the enumeration problems of knots of given genus, and the sl_N weight system of trivalent graphs known from the theory of Vassiliev invariants. One can express the maximal volume of knots of given canonical genus by links L_G associated to planar trivalent graphs G similarly to Habiro's claspers. The sl_N weight system of G then is related to both the hyperbolic volume and the enumeration of knots by genus, and also to the enumeration of 1-vertex triangulations of oriented surfaces carried out by my collaborator A. Vdovina. Weight system-volume-conjectures The form of the relation between the sl_N weight system W_N of G and volume of L_G is not yet clear, but calculations suggest that definitely something is going on beyond accidental coincidences. I hope in the future to deepen my understanding of hyperbolic volumes, in particular to understand better these relations. I hope also to find out whether and what is a relation of these coincidences to the Volume conjecture. For example, can one calculate the colored Jones polynomials of L_G, and establish the relation modulo the Volume conjecture? Can one gain insight into the Volume conjecture from these relations? This is also linked to understanding the, in particular multiplicative, structure of the sl_N weight systems of G. Few facts are known, including the multiplicative character of Vogel's algebra and Bar-Natan's version of the 4-Color-Theorem. A new observation from the work of Bacher and Vdovina on 1-vertex triangulations is that the linear term of the sl_N weight systems vanishes in Euler characteristic <-1. Their work also implies bounds on the number of linear monomials in the calculation of W_N, that in turn bound the asymptotical growth of the number of alternating knots of given genus, which I seek to improve. I hope to progress on at least some of these many interrelated problems in the future. Gauss sum invariants There are still opportunities left in the application of Gauss sum invariants to positive knots and related knot classes, most naturally, to improve the existing inequalities. More importantly, a computational project is to implement Fiedler's new character Gauss sum invariants (which take as input not a single diagram, but a sequence of diagrams of knots in the solid torus), in the hope to distinguish knot orientation with them, after the success on braids. This will obviously give a huge impetus on the theory of Gauss sum invariants. Number theoretic properties of knot invariants One of my original mathematical interests was number theory (my specialization turned into a different direction by the influence I experienced during my studies). I'm interested in situations in which knot invariants can be studied from the point of view of some elementary number theoretic properties. A series of problems I intend to work on is related to determinants of achiral knots with particular properties, for example unknotting number one. These determinants are sums of two squares. It would be interesting to study which such numbers occur in which situations. Number theoretic properties of the determinant have also applications to unknotting numbers and knot distance, and maybe I can find more such applications. Non-trivial Jones polynomial problem I seek further generalizations of the non-triviality result for the Jones polynomial, for example to arborescent knots. I also try to prove that there are infinitely many positive knots with no positive minimal crossing diagrams (a problem raised by Nakamura), and achiral knots of any odd crossing number at least 15. Other topics I'm also working on some problems of braids, for example the question of Rudolph whether (strongly) quasipositive knots have (strongly) quasipositive braid representations of minimal strand number. I expect counterexamples but they are not easy to construct. (I had in the past some counterexamples for positive braid representations [St18] and, jointly with M. Hirasawa [HS], minimal genus band representations.) A more expanded version of my research summary and plan may be found at http://www.ms.u-tokyo.ac.jp/~stoimeno/tpl.ps.gz SELECTED PUBLICATIONS [BS] The Fundamental Theorem of Vassiliev invariants, joint with D. Bar-Natan, "Geometry and Physics", Lecture Notes in Pure & Appl. Math. 184, M. Dekker, New York, 1996, 101--134. [St] Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Of Knot Theory and Its Ram. 7(1) (1998), 93--114. [St2] Stirling numbers, Eulerian idempotents and a diagram complex, J. Of Knot Theory and Its Ram. 7(2) (1998), 231--256. [St3] Gauss sum invariants, Vassiliev invariants and braiding sequences, J. Of Knot Theory and Its Ram. 9(2) (2000), 221--269. [St4] On finiteness of Vassiliev invariants and a proof of the Lin-Wang conjecture via braiding polynomials, J. Of Knot Theory and Its Ram. 10(5) (2001), special volume for the proceedings of the International Conference on Knot Theory "Knots in Hellas, 98", 769--780. [St5] On the number of chord diagrams, Discr. Math. 218 (2000), 209--233. [St6] Positive knots, closed braids and the Jones polynomial, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237--285. [St7] Mutant links distinguished by degree 3 Gauss sums, Proceedings of the International Conference on Knot Theory "Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000. [St8] Genera of knots and Vassiliev invariants, J. Of Knot Theory and Its Ram. 8(2) (1999), 253--259. [FS] New knot and link invariants, joint with T. Fiedler, Proceedings of the International Conference on Knot Theory "Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000. [St9] Gauss sums on almost positive knots, Compositio Mathematica 140(1) (2004), 228--254. [St10] Knots of genus one, Proc. Amer. Math. Soc. 129(7) (2001), 2141--2156. [St11] The Conway Vassiliev invariants on twist knots, Kobe J. Math. 16(2) (1999), 189--193. [St12] Vassiliev invariants and rational knots of unknotting number one, Topology 42(1) (2003), 227--241. [KS] The crossing number and maximal bridge length of a knot diagram, with an appendix by M. Kidwell, Pacific J. Math. 210(1) (2003), 189--199. [St13] Branched cover homology and Q evaluations, Osaka J. Math. 39(1) (2002), 13--21. [St14] Rational knots and a theorem of Kanenobu, Exper. Math. 9(3) (2000), 473--478. [St15] Fibonacci numbers and the 'fibered' Bleiler conjecture, Int. Math. Res. Notices 23 (2000), 1207--1212. [St16] Some examples related to 4-genera, unknotting numbers, and knot polynomials, Jour. London Math. Soc. 63(2) (2001), 487--500. [St17] Some inequalities between knot invariants, Internat. J. Math. 13(4) (2002), 373--393. [St18] On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks, Trans. Amer. Math. Soc. 354 (10) (2002), 3927--3954. [STV] The canonical genus of a classical and virtual knot, joint with V. Tchernov and A. Vdovina, Geometriae Dedicata 95(1) (2002), 215--225. [St19] On the number of links and link polynomials, Quart. J. Math. Oxford. 55(1) (2004), 87-98. [St20] The skein polynomial of closed 3-braids, J. Reine Angew. Math. 564 (2003), 167--180. [HS] Examples of knots without minimal string Bennequin surfaces, joint with M. Hirasawa, Asian Journal of Mathematics 7(3) (2003), 435--446. [MS] The Alexander polynomial of planar even valence graphs, joint with K. Murasugi, Adv. Appl. Math. 31(2) (2003), 440--462.