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Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

Dror Bar-Natan and Zsuzsanna Dancso

Algebraic & Geometric Topology 16 (2016) 1063–1133
1 A Alekseev, B Enriquez, C Torossian, Drinfeld associators, braid groups and explicit solutions of the Kashiwara–Vergne equations, Publ. Math. Inst. Hautes Études Sci. (2010) 143 MR2737979
2 A Alekseev, C Torossian, The Kashiwara–Vergne conjecture and Drinfeld’s associators, Ann. of Math. 175 (2012) 415 MR2877064
3 Z Amir-Khosravi, S Sankaran, VasCalc: a Vassiliev invariants calculator,
4 E Artin, Theory of braids, Ann. of Math. 48 (1947) 101 MR0019087
5 J C Baez, D K Wise, A S Crans, Exotic statistics for strings in 4D BF theory, Adv. Theor. Math. Phys. 11 (2007) 707 MR2362007
6 D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423 MR1318886
7 D Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995) 13 MR1321289
8 D Bar-Natan, Some computations related to Vassiliev invariants, (1996)
9 D Bar-Natan, Vassiliev and quantum invariants of braids, from: "The interface of knots and physics" (editor L H Kauffman), Proc. Sympos. Appl. Math. 51, Amer. Math. Soc. (1996) 129 MR1372767
10 D Bar-Natan, Non-associative tangles, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 139 MR1470726
11 D Bar-Natan, On associators and the Grothendieck–Teichmuller group, I, Selecta Math. 4 (1998) 183 MR1669949
12 D Bar-Natan, Algebraic knot theory —a call for action, web document (2006)
13 D Bar-Natan, Finite-type invariants, from: "Encyclopedia of Mathematical Physics" (editors J P Françoise, G L Naber, T S Tsun), Elsevier (2006)
14 D Bar-Natan, Balloons and hoops and their universal finite-type invariant, BF theory, and an ultimate Alexander invariant, Acta Math. Vietnam. 40 (2015) 271 MR3366171
15 D Bar-Natan, Z Dancso, Finite type invariants of w-knotted objects: from Alexander to Kashiwara and Vergne, web version of the first and second papers in one, videos (wClips) and related files
16 D Bar-Natan, Z Dancso, Finite type invariants of w-knotted objects, II: Tangles and the Kashiwara-Vergne problem, preprint (2014) arXiv:1405.1955
17 D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres, I : A highly non trivial flat connection on S3, Selecta Math. 8 (2002) 315 MR1931167
18 D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres, II : Invariance and universality, Selecta Math. 8 (2002) 341 MR1931168
19 D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The Århus integral of rational homology 3–spheres, III : Relation with the Le–Murakami–Ohtsuki invariant, Selecta Math. 10 (2004) 305 MR2099069
20 D Bar-Natan, I Halacheva, L Leung, F Roukema, Some dimensions of spaces of finite type invariants of virtual knots, Exp. Math. 20 (2011) 282 MR2836253
21 D Bar-Natan, T T Q Le, D P Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003) 1 MR1988280
22 D Bar-Natan, S Selmani, Meta-monoids, meta-bicrossed products, and the Alexander polynomial, J. Knot Theory Ramifications 22 (2013) MR3125897
23 D Bar-Natan, A Stoimenow, The fundamental theorem of Vassiliev invariants, from: "Geometry and physics" (editors J E Andersen, J Dupont, H Pedersen, A Swann), Lecture Notes in Pure and Appl. Math. 184, Dekker (1997) 101 MR1423158
24 V G Bardakov, The virtual and universal braids, Fund. Math. 184 (2004) 1 MR2128039
25 V G Bardakov, P Bellingeri, Combinatorial properties of virtual braids, Topology Appl. 156 (2009) 1071 MR2493369
26 B Berceanu, Ş Papadima, Universal representations of braid and braid-permutation groups, J. Knot Theory Ramifications 18 (2009) 999 MR2549480
27 R Bott, C Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994) 5247 MR1295465
28 T E Brendle, A Hatcher, Configuration spaces of rings and wickets, Comment. Math. Helv. 88 (2013) 131 MR3008915
29 J S Carter, M Saito, Knotted surfaces and their diagrams, 55, Amer. Math. Soc. (1998) MR1487374
30 A S Cattaneo, P Cotta-Ramusino, J Fröhlich, M Martellini, Topological BF theories in 3 and 4 dimensions, J. Math. Phys. 36 (1995) 6137 MR1355902
31 A S Cattaneo, P Cotta-Ramusino, M Martellini, Three-dimensional BF theories and the Alexander–Conway invariant of knots, Nuclear Phys. B 436 (1995) 355 MR1316372
32 Z Dancso, On the Kontsevich integral for knotted trivalent graphs, Algebr. Geom. Topol. 10 (2010) 1317 MR2661529
33 V G Drinfel’d, Quantum groups, from: "Proceedings of the International Congress of Mathematicians, I, II" (editor A M Gleason), Amer. Math. Soc. (1987) 798 MR934283
34 V G Drinfel’d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989) 114 MR1047964
35 V G Drinfel’d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q∕Q), Algebra i Analiz 2 (1990) 149 MR1080203
36 H A Dye, Virtual knots undetected by 1– and 2–strand bracket polynomials, Topology Appl. 153 (2005) 141 MR2172041
37 D B A Epstein, Word processing in groups, A K Peters (1992)
38 P Etingof, D Kazhdan, Quantization of Lie bialgebras, I, Selecta Math. 2 (1996) 1 MR1403351
39 R Fenn, R Rimányi, C Rourke, The braid-permutation group, Topology 36 (1997) 123 MR1410467
40 D L Goldsmith, The theory of motion groups, Michigan Math. J. 28 (1981) 3 MR600411
41 M Goussarov, Finite type invariants and n–equivalence of 3–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517 MR1715131
42 M Goussarov, M Polyak, O Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045 MR1763963
43 M Gusarov, On n–equivalence of knots and invariants of finite degree, from: "Topology of manifolds and varieties" (editor O Viro), Adv. Soviet Math. 18, Amer. Math. Soc. (1994) 173 MR1296895
44 M Gutiérrez, S Krstić, Normal forms for basis-conjugating automorphisms of a free group, Internat. J. Algebra Comput. 8 (1998) 631 MR1682236
45 N Habegger, G Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000) 1253 MR1783857
46 K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 MR1735632
47 K Habiro, T Kanenobu, A Shima, Finite type invariants of ribbon 2–knots, from: "Low-dimensional topology" (editor H Nencka), Contemp. Math. 233, Amer. Math. Soc. (1999) 187 MR1701683
48 K Habiro, A Shima, Finite type invariants of ribbon 2–knots, II, Topology Appl. 111 (2001) 265 MR1814229
49 A Haviv, Towards a diagrammatic analogue of the Reshetikhin–Turaev link invariants, PhD thesis, Hebrew University (2002) arXiv:math.QA/0211031
50 D Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37 MR638121
51 T Kanenobu, A Shima, Two filtrations of ribbon 2–knots, Topology Appl. 121 (2002) 143 MR1903688
52 M Kashiwara, M Vergne, The Campbell–Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978) 249 MR0492078
53 L H Kauffman, On knots, 115, Princeton Univ. Press (1987) MR907872
54 L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663 MR1721925
55 L H Kauffman, S Lambropoulou, Virtual braids, Fund. Math. 184 (2004) 159 MR2128049
56 J A Kneissler, The number of primitive Vassiliev invariants up to degree twelve, preprint (1997) arXiv:q-alg/9706022
57 T Kohno, Vassiliev invariants and de Rham complex on the space of knots, from: "Symplectic geometry and quantization" (editors Y Maeda, H Omori, A Weinstein), Contemp. Math. 179, Amer. Math. Soc. (1994) 123 MR1319605
58 A Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture, preprint (2000) arXiv:math/0005284
59 G Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003) 587 MR1997331
60 V Kurlin, Compressed Drinfeld associators, Journal of Algebra 292 (2005) 184 MR2166802
61 T T Q Le, An invariant of integral homology 3–spheres which is universal for all finite type invariants, from: "Solitons, geometry, and topology: on the crossroad" (editors V M Buchstaber, S P Novikov), Amer. Math. Soc. Transl. Ser. 2 179 (1997) 75
62 T Q T Le, J Murakami, Kontsevich’s integral for the HOMFLY polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995) 193 MR1320252
63 T Q T Le, J Murakami, The universal Vassiliev–Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996) 41 MR1394520
64 T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3–manifolds, Topology 37 (1998) 539 MR1604883
65 P Lee, Closed-form associators and braidors in a partly commutative quotient, preprint (2007)
66 P Lee, The pure virtual braid group is quadratic, Selecta Math. 19 (2013) 461 MR3090235
67 P Lee, Proof of a conjectured formula for the Alexander invariant, J. Knot Theory Ramifications 23 (2014) MR3298210
68 C Lescop, On the cube of the equivariant linking pairing for 3–manifolds of rank one, preprint arXiv:1008.5026
69 C Lescop, Invariants of knots and 3–manifolds derived from the equivariant linking pairing, from: "Chern–Simons gauge theory : 20 years after" (editors J E Andersen, H U Boden, A Hahn, B Himpel), AMS/IP Stud. Adv. Math. 50, Amer. Math. Soc. (2011) 217 MR2809454
70 L Leung, Combinatorial formulas for classical Lie weight systems on arrow diagrams, preprint (2008) arXiv:0812.2342
71 J Lieberum, The Drinfeld associator of gl(1|1), from: "Quantum groups" (editor B Enriquez), IRMA Lect. Math. Theor. Phys. 12, Eur. Math. Soc. (2008) 39 MR2432989
72 X S Lin, Power series expansions and invariants of links, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 184 MR1470727
73 W Magnus, A Karrass, D Solitar, Combinatorial group theory : Presentations of groups in terms of generators and relations, Interscience (1966) MR0207802
74 J McCool, On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (1986) 1525 MR873421
75 J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211 MR0174052
76 G Naot, On Chern–Simons theory with an inhomogeneous gauge group and BF theory knot invariants, J. Math. Phys. 46 (2005) MR2194021
77 T Ohtsuki, Finite type invariants of integral homology 3–spheres, J. Knot Theory Ramifications 5 (1996) 101 MR1373813
78 M Polyak, On the algebra of arrow diagrams, Lett. Math. Phys. 51 (2000) 275 MR1778064
79 D Rolfsen, Knots and links, 7, Publish or Perish (1990) MR1277811
80 F Roukema, Goussarov–Polyak–Viro combinatorial formulas for finite type invariants, preprint (2007) arXiv:0711.4001
81 S Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications 9 (2000) 531 MR1758871
82 D Thurston, Integral expressions for the Vassiliev knot invariants, senior thesis, Harvard University (1995) arXiv:math.QA/9901110
83 V A Vassiliev, Cohomology of knot spaces, from: "Theory of singularities and its applications" (editor V I Arnol’d), Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 23 MR1089670
84 T Watanabe, Clasper-moves among ribbon 2–knots characterizing their finite type invariants, J. Knot Theory Ramifications 15 (2006) 1163 MR2287439