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Problems on Invariants of Knots and 3-Manifolds Preface

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Problems on Invariants of Knots and 3-Manifolds Preface i Problems on invariants of knots and 3-manifolds Edited by T. Ohtsuki Preface The workshop and seminars on “Invariants of knots and 3-manifolds” was held at Research Institute for Mathematical Sciences, Kyoto University in September 2001. There were 25 talks in the workshop in September 17–21, and there were 27 talks in the seminars in the other weeks of September. Each speaker was requested to give his/her open problems in a short problem session after his/her talk, and many interesting open problems were given and discussed by the speakers and participants in the workshop and the seminars. Contributors of the open problems were also requested to give kind expositions of history, background, significance, and/or importance of the problems. This problem list was made by editing these open problems and such expositions.1 Since the interaction between geometry and mathematical physics in the 1980s, many invariants of knots and 3-manifolds have been discovered and studied. The discovery and analysis of the enormous number of these invariants yielded a new area: the study of invariants of knots and 3-manifolds (from another viewpoint, the study of the sets of knots and 3-manifolds). Recent works have almost completed the topological reconstruction of the invariants derived from the Chern-Simons field theory, which was one of main problems of this area. Further, relations among these invariants have been studied enough well, and these invariants are now well-organized. For the future developments of this area, it might be important to consider various streams of new directions;2 this is a reason why the editor tried to make the problem list expository. The editor hopes this problem list will clarify the present frontier of this area and assist readers when considering future directions. The editor will try to keep up-to-date information on this problem list at his web site.3 If the reader knows a (partial) solution of any problem in this list, please let him4 know it. February, 2003 T. Ohtsuki The logo for the workshop and the seminars was designed by N. Okuda. 1Open problems on the Rozansky-Witten invariant were written in a separate manuscript [349]. Some fundamental problems are quoted from other problem lists such as [188], [220], [262], [285], [286], [388, Pages 571–572]. 2For example, directions related to other areas such as hyperbolic geometry via the volume conjecture and the theory of operator algebras via invariants arising from 6j-symbols. 3http://www.kurims.kyoto-u.ac.jp/~tomotada/proj01/ 4Email address of the editor is: [email protected] eometry & opology onographs, Volume 4 (2002) G T M ii Contents Preface 1 0 Introduction 377 1 Polynomial invariants of knots 380 1.1 TheJonespolynomial ............................ 380 1.1.1 Does the Jones polynomial distinguish the trivial knot? ..... 381 1.1.2 Characterization and interpretation of the Jones polynomial . 382 1.1.3 Numericalexperiments. 387 1.1.4 Categorificationof the Jones polynomial . 390 1.2 TheHOMFLY,Q,andKauffmanpolynomials. 390 1.3 Thevolumeconjecture.. .. .. .. .. .. .. .. .. .. .. .. 393 2 Finite type invariants of knots 398 2.1 TorsionandVassilievinvariants . 399 2.2 Do Vassiliev invariants distinguish knots? . ....... 401 2.3 Can Vassiliev invariants detect other invariants? . .........402 2.4 Vassiliev invariants and crossing numbers . ...... 403 2.5 Dimensions of spaces of Vassiliev invariants . ....... 405 2.6 Milnorinvariants ............................... 408 2.7 Finite type invariants of virtual knots . .. .. 409 2.8 Finite type invariants derived from local moves . ....... 412 2.9 Loopfinitetypeinvariants. 415 2.10 Goussarov-Habirotheoryforknots . 418 2.11Otherproblems................................ 422 3 The Kontsevich invariant 424 3.1 Calculation of the Kontsevich invariant . .. .. 424 3.2 Does the Kontsevich invariant distinguish knots? . ........425 3.3 Characterization and interpretation of the Kontsevich invariant . 425 3.4 The Kontsevich invariant in a finite field . 426 3.5 The Kontsevichinvariant in arrow diagrams . .. .. 427 3.6 The Chern-Simons series of configuration space integrals.........429 3.7 Associators ..................................432 3.8 Graphcohomology .............................. 435 3.9 The loop expansion of the Kontsevich invariant . ...... 437 3.10 The Kontsevich invariant of links in Σ [0, 1]...............443 × eometry & opology onographs, Volume 4 (2002) G T M iii 4 Skein modules 445 4.1 TheKauffmanbracketskeinmodule . 446 4.2 TheHomflyptskeinmodule . 447 4.3 TheKauffmanskeinmodule. 448 4.4 The q-homotopyskeinmodule . 449 4.5 The (4, )skeinmodule........................... 450 ∞ 4.6 Otherproblems................................ 453 5 Quandles 455 5.1 Classificationofquandles . 455 5.2 Representationsofknotquandles . 458 5.3 (Co)homologyofquandles . 459 5.4 Quandlecocycleinvariant . 461 5.5 Quantumquandles .............................. 464 5.6 Rack(co)homology.............................. 465 6 Braid group representations 466 6.1 TheTemperley-Liebalgebra. 466 6.2 TheBuraurepresentation . 467 6.3 TheHeckeandBMWalgebras . 468 6.4 Otherproblems................................ 469 7 Quantum and perturbative invariants of 3-manifolds 471 7.1 Witten-Reshetikhin-Turaev invariants and quantum invariants . 471 7.2 Theasymptoticexpansionconjecture. 474 7.3 Thevolumeconjecture.. .. .. .. .. .. .. .. .. .. .. .. 481 7.4 Quantum hyperbolic invariants of 3-manifolds . ....... 485 7.5 Perturbativeinvariants. 488 8 Topological quantum field theory 493 8.1 Classification and characterization of TQFT’s . ....... 494 8.2 SpinTQFT’s .................................495 8.3 HomotopyQFT’s............................... 496 8.4 GeometricconstructionofTQFT’s . 496 8.5 Half-projective andhomologicalTQFT’s . .. .. 498 9 The state-sum invariants of 3-manifolds derived from 6j -symbols 506 9.1 Monoidal categories, 6j -symbols,andsubfactors . 506 9.2 Turaev-Viro invariants and the state-sum invariants derived from monoidal categories508 eometry & opology onographs, Volume 4 (2002) G T M iv 9.3 The state-sum invariants derived from ribbon categories .........511 9.4 Turaev-Viro-Ocneanuinvariants. 512 10 Casson invariant and finite type invariants of 3-manifolds 514 10.1 CassonandRokhlininvariants . 514 10.2 Finitetypeinvariants . 517 10.2.1 Torsionandfinitetypeinvariants . 518 10.2.2 Do finite type invariants distinguish homology 3-spheres? . 518 10.2.3 Dimensions of spaces of finite type invariants . .. .. 518 10.2.4 Vogel’salgebra . 520 10.2.5 Otherproblems. 522 10.3 Goussarov-Habirotheory. 523 10.3.1 Goussarov-Habiro theory for 3-manifolds . .. .. 523 10.3.2 Goussarov-Habiro theory for spin and spinc 3-manifolds . 525 11 The LMO invariant 527 11.1 CalculationoftheLMOinvariant. 527 11.2 Does the LMO invariant distinguish integral homology 3-spheres? . 527 11.3 Characterization of the image of the LMO invariant . ....... 528 11.4 VariationsoftheLMOinvariant . 528 11.5 RefinementsoftheLMOinvariant . 528 11.6Otherproblems................................ 529 12 Other problems 530 12.1 (Pseudo)Legendrianknotinvariants . .. .. 530 12.2 Knotsandfinitegroups . .. .. .. .. .. .. .. .. .. .. .. 531 12.3 The numbers of 3-, 5-colorings and some local moves . ....... 532 12.4 Knottedtrivalentgraphs. 533 12.5Quantumgroups ............................... 537 12.6Otherproblems................................ 538 Bibliography 543 Index of contributors 567 Index 568 eometry & opology onographs, Volume 4 (2002) G T M ISSN 1464-8997 (on line) 1464-8989 (printed) 377 eometry & opology onographs G T M Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001) Pages 377–572 Problems on invariants of knots and 3-manifolds Edited by T. Ohtsuki Abstract This is a list of open problems on invariants of knots and 3- manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in prob- lem sessions in the workshop and seminars on “Invariants of Knots and 3-Manifolds” held at Kyoto in 2001. AMS Classification 20F36, 57M25, 57M27, 57R56; 13B25, 17B10, 17B37, 18D10, 20C08, 20G42, 22E99, 41A60, 46L37, 57M05, 57M50, 57N10, 57Q10, 81T18, 81T45 Keywords Invariant, knot, 3-manifold, Jones polynomial, Vassiliev in- variant, Kontsevich invariant, skein module, quandle, braid group, quantum invariant, perturbative invariant, topological quantum field theory, state- sum invariant, Casson invariant, finite type invariant, LMO invariant 0 Introduction The study of quantum invariants of links and three-manifolds has a strange status within topology. When it was born, with Jones’ 1984 discovery of his famous polynomial [186], it seemed that the novelty and power of the new invariant would be a wonderful tool with which to resolve some outstanding questions of three-dimensional topology. Over the last 16 years, such hopes have been largely unfulfilled, the only obvious exception being the solution of the Tait conjectures about alternating knots (see for example [281]). This is a disappointment, and particularly so if one expects the role of the quan- tum invariants in mathematics to be the same as that of the classical invariants of three-dimensional topology. Such a comparison misses the point that most of the classical invariants were created specifically in order to distinguish between things; their definitions are mainly intrinsic, and it is therefore clear what kind of topological properties they reflect, and how to attempt to use them to solve topological problems. Chapter 0 was written by J. Roberts. Published
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