A geometrical presentation of the surface mapping class group and surgery

Matveev, Sergei; Polyak, Michael

doi:10.1007/bf02173428

Cited by 20 publications

(24 citation statements)

References 11 publications

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“…, g and e 2 [16,12]. Knowing the geometrical realization of M g,1 by the isomorphic group, T 2g of certain equivalence classes of admissible tangles [12], a central extensionM g,1 :=T 2g of M g,1 can be easily obtained by splitting up equivalence classes in T 2g : the central generator of z, which has been the K 1 -equivalence in [12], will correspond to an insertion (elimination) of a ±1 (∓1) framed separated unknot into (from) a tangle diagram. On the level of tangles it clearly defines a central generator, which in terms of a RHA is represented by multiplication by the phase exp(±2πic/8).…”

Section: Unitary Representations Ofm G1mentioning

confidence: 99%

Mapping Class Group Representations and Generalized Verlinde Formula

Bantay

Vecsernyés

1999

Int. J. Mod. Phys. A

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Section: Unitary Representations Ofm G1mentioning

confidence: 99%

Mapping Class Group Representations and Generalized Verlinde Formula

Bantay

Vecsernyés

1999

Int. J. Mod. Phys. A

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“…and K 2 (see [9] for K 4 , K 5 , and [11] for K 3 ): K 4 : Delete a ±1-framed unknot, at the expense of the full left-or right-hand twist on the strings linked with it.…”

Section: Surgery On Linksmentioning

confidence: 99%

Calculus of clovers and finite type invariants of 3–manifolds

Goussarov²,

2001

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A clover is a framed trivalent graph with some additional structure, embedded in a 3-manifold. We define surgery on clovers, generalizing surgery on Y-graphs used earlier by the second author to define a new theory of finite-type invariants of 3-manifolds. We give a systematic exposition of a topological calculus of clovers and use it to deduce some important results about the corresponding theory of finite type invariants. In particular, we give a description of the weight systems in terms of uni-trivalent graphs modulo the AS and IHX relations, reminiscent of the similar results for links. We then compare several definitions of finite type invariants of homology spheres (based on surgery on Y-graphs, blinks, algebraically split links, and boundary links) and prove in a self-contained way their equivalence.

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“…For our generating set of Dehn twists η νp , α i , β i , δ i , ǫ i the corresponding diagrams can be found e.g. in [35]. Then the elements in the diagram are represented by objects associated with a Hopf algebra G in the usual way (cp.…”

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confidence: 99%

Representation theory of Chern-Simons observables

Alekseev¹,

Schomerus²

1996

Duke Math. J.

151

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In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian Chern-Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.HUTMP 95-B342 ETH-TH/ 95-12 ESI 214 (1995) hep-th 9503016 * On leave of absence from Steklov Mathematical Institute,

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A geometrical presentation of the surface mapping class group and surgery

Abstract: We construct a tangle presentation of the mapping class group similar to a natural presentation of the braid group by geometrical braids. A relation between surgery and Heegard diagrams for 3-manifolds arising in this way and different applications are studied.

Cited by 20 publications

References 11 publications

Mapping Class Group Representations and Generalized Verlinde Formula

Mapping Class Group Representations and Generalized Verlinde Formula

Calculus of clovers and finite type invariants of 3–manifolds

Representation theory of Chern-Simons observables

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