Geodesic
An invariant of link cobordisms from Khovanov homology
Jacobsson, Magnus1
1Istituto Nazionale di Alta Matematica (INdAM), Città Universitaria, P.le Aldo Moro 5, 00185 Roma, Italy
Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1211-1251
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In [Duke Math. J. 101 (1999) 359–426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lefschetz polynomial of the trivial cobordism is the Jones polynomial. These polynomials can be computed on the chain level.

DOI : 10.2140/agt.2004.4.1211
Keywords: Khovanov homology, link cobordism, Jones polynomial

Jacobsson, Magnus  1

1 Istituto Nazionale di Alta Matematica (INdAM), Città Universitaria, P.le Aldo Moro 5, 00185 Roma, Italy 
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Jacobsson, Magnus. An invariant of link cobordisms from Khovanov homology. Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1211-1251. doi: 10.2140/agt.2004.4.1211

[1] L Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996) 569

[2] D Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337

[3] J C Baez, L Langford, Higher-dimensional algebra IV: 2–tangles, Adv. Math. 180 (2003) 705

[4] J S Carter, M Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies, J. Knot Theory Ramifications 2 (1993) 251

[5] J S Carter, D Jelsovsky, S Kamada, L Langford, M Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003) 3947

[6] J S Carter, J H Rieger, M Saito, A combinatorial description of knotted surfaces and their isotopies, Adv. Math. 127 (1997) 1

[7] V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. $($N.S.$)$ 12 (1985) 103

[8] M Jacobsson, An invariant of link cobordisms from Khovanov's homology theory, pre-publication version

[9] M Jacobsson, Khovanov's conjecture over $\mathbb{Z}[c]$

[10] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359

[11] L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395

[12] D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: "Knot theory (Warsaw, 1995)", Banach Center Publ. 42, Polish Acad. Sci. (1998) 347

[13] O Viro, Khovanov homology, its definitions and ramifications, Fund. Math. 184 (2004) 317

[14] O P Östlund, personal communication

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