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Claspers and finite type invariants of links. (English) Zbl 0941.57015

This landmark paper introduces the theory of claspers. The same theory was independently introduced by M. N. Gusarov [St. Petersbg. Math. J. 12, No. 4, 569–604 (2001); translation from Algebra Anal. 12, No. 4, 79–125 (2001; Zbl 0981.57006); C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 517–522 (1999: Zbl 0938.57013)], S. Garoufalidis, M. Goussarov and M. Polyak [Geom. Topol. 5, 75–108 (2001; Zbl 1066.57015)].
Clasper theory serves a number of purposes. First and foremost, it provides a topological counterpart to Jacobi diagrams, explaining their Lie algebra structure in terms of the natural Hopf algebraic structure of a certain category of cobordisms. Secondly, for each positive integer \(n\), claspers provide a natural set of local moves (\(C_n\)-moves) which modify knots while preserving their invariants of type \(\leq n\). Third, claspers provide a convenient graphical language with which to manipulate knotted objects, whose formalism makes various classical arguments (for example the proof that any two links with the same linking matrix are related by \(\Delta\)–moves [H. Murakami, Y. Nakanishi, Math. Ann. 284, No. 1, 75–89 (1989; Zbl 0646.57005)]) look very easy. Fourth, claspers give rise to algebraically convenient surgery presentations of knots and of boundary links.
This is one of the most important papers in quantum topology.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
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References:

[1] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423 · Zbl 0898.57001 · doi:10.1016/0040-9383(95)93237-2
[2] D Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995) 13 · Zbl 0878.57003 · doi:10.1142/S021821659500003X
[3] J S Birman, New points of view in knot theory, Bull. Amer. Math. Soc. \((\)N.S.\()\) 28 (1993) 253 · Zbl 0785.57001 · doi:10.1090/S0273-0979-1993-00389-6
[4] J S Birman, X S Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993) 225 · Zbl 0812.57011 · doi:10.1007/BF01231287
[5] T D Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990) · Zbl 0705.57003
[6] T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on 3-manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97 · Zbl 0984.57010 · doi:10.1017/S0305004101005151
[7] T D Cochran, P Melvin, Finite type invariants of 3-manifolds, Invent. Math. 140 (2000) 45 · Zbl 0949.57010 · doi:10.1007/s002220000045
[8] L Crane, D Yetter, On algebraic structures implicit in topological quantum field theories, J. Knot Theory Ramifications 8 (1999) 125 · Zbl 0935.57025 · doi:10.1142/S0218216599000109
[9] M H Freedman, P Teichner, 4-manifold topology II: Dwyer’s filtration and surgery kernels, Invent. Math. 122 (1995) 531 · Zbl 0857.57018 · doi:10.1007/BF01231455
[10] S Garoufalidis, N Habegger, The Alexander polynomial and finite type 3-manifold invariants, Math. Ann. 316 (2000) 485 · Zbl 0952.57002 · doi:10.1007/s002080050340
[11] S Garoufalidis, J Levine, Finite type 3-manifold invariants, the mapping class group and blinks, J. Differential Geom. 47 (1997) 257 · Zbl 0917.57009
[12] S Garoufalidis, T Ohtsuki, On finite type 3-manifold invariants III: Manifold weight systems, Topology 37 (1998) 227 · Zbl 0889.57017 · doi:10.1016/S0040-9383(97)00028-1
[13] M N Goussarov, A new form of the Conway-Jones polynomial of oriented links, Adv. Soviet Math. 18, Amer. Math. Soc. (1994) 167 · Zbl 0816.57006
[14] M N Goussarov, On \(n\)-equivalence of knots and invariants of finite degree, Adv. Soviet Math. 18, Amer. Math. Soc. (1994) 173 · Zbl 0865.57007
[15] M N Goussarov, Interdependent modifications of links and invariants of finite degree, Topology 37 (1998) 595 · Zbl 0949.57004 · doi:10.1016/S0040-9383(97)00026-8
[16] M N Goussarov, New theory of invariants of finite degree for 3-manifolds, preprint · Zbl 0938.57013
[17] N Habegger, A computation of the universal quantum 3-manifold invariant for manifolds of rank greater than 2, preprint
[18] N Habegger, X S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389 · Zbl 0704.57016 · doi:10.2307/1990959
[19] N Habegger, G Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000) 1253 · Zbl 0964.57011 · doi:10.1016/S0040-9383(99)00041-5
[20] K Habiro, Claspers and the Vassiliev skein modules, PhD thesis, University of Tokyo (1997)
[21] K Habiro, Clasp-pass moves on knots, unpublished
[22] R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597 · Zbl 0915.57001 · doi:10.1090/S0894-0347-97-00235-X
[23] D Johnson, An abelian quotient of the mapping class group \(\mathcalI_g\), Math. Ann. 249 (1980) 225 · Zbl 0409.57009 · doi:10.1007/BF01363897
[24] T Kerler, Genealogy of non-perturbative quantum-invariants of 3-manifolds: the surgical family, Lecture Notes in Pure and Appl. Math. 184, Dekker (1997) 503 · Zbl 0869.57014
[25] T Kerler, Bridged links and tangle presentations of cobordism categories, Adv. Math. 141 (1999) 207 · Zbl 0937.57017 · doi:10.1006/aima.1998.1772
[26] R Kirby, A calculus for framed links in \(S^3\), Invent. Math. 45 (1978) 35 · Zbl 0377.55001 · doi:10.1007/BF01406222
[27] T Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179, Amer. Math. Soc. (1994) 123 · Zbl 0876.57009
[28] M Kontsevich, Vassiliev’s knot invariants, Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 137 · Zbl 0839.57006
[29] V S Krushkal, Additivity properties of Milnor’s \(\bar{\mu}\)-invariants, J. Knot Theory Ramifications 7 (1998) 625 · Zbl 0931.57005 · doi:10.1142/S0218216598000322
[30] T T Q Le, An invariant of integral homology 3-spheres which is universal for all finite type invariants, Amer. Math. Soc. Transl. Ser. 2 179, Amer. Math. Soc. (1997) 75 · Zbl 0914.57013
[31] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3-manifolds, Topology 37 (1998) 539 · Zbl 0897.57017 · doi:10.1016/S0040-9383(97)00035-9
[32] X S Lin, Power series expansions and invariants of links, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 184 · Zbl 0897.57006
[33] S Majid, Algebras and Hopf algebras in braided categories, Lecture Notes in Pure and Appl. Math. 158, Dekker (1994) 55 · Zbl 0812.18004
[34] S Majid, Foundations of quantum group theory, Cambridge University Press (1995) · Zbl 0857.17009
[35] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345 · Zbl 0634.57006
[36] J Milnor, Link groups, Ann. of Math. \((2)\) 59 (1954) 177 · Zbl 0055.16901 · doi:10.2307/1969685
[37] J Milnor, Isotopy of links. Algebraic geometry and topology, Princeton University Press (1957) 280 · Zbl 0080.16901
[38] S Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I, Topology 28 (1989) 305 · Zbl 0684.57008 · doi:10.1016/0040-9383(89)90011-6
[39] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699 · Zbl 0801.57011 · doi:10.1215/S0012-7094-93-07017-2
[40] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75 · Zbl 0646.57005 · doi:10.1007/BF01443506
[41] K Y Ng, Groups of ribbon knots, Topology 37 (1998) 441 · Zbl 0887.57010 · doi:10.1016/S0040-9383(97)00037-2
[42] T Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications 5 (1996) 101 · Zbl 0942.57009 · doi:10.1142/S0218216596000084
[43] T Stanford, Finite-type invariants of knots, links, and graphs, Topology 35 (1996) 1027 · Zbl 0863.57005 · doi:10.1016/0040-9383(95)00056-9
[44] T Stanford, Braid commutators and Vassiliev invariants, Pacific J. Math. 174 (1996) 269 · Zbl 0868.57016
[45] T Stanford, Vassiliev invariants and knots modulo pure braid subgroups, preprint
[46] V A Vassiliev, Cohomology of knot spaces, Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 23 · Zbl 1015.57003
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