Geodesic
The state sum invariant of 3–manifolds constructed from the E6 linear skein
Okazaki, Kenta1
1Research Institute for Mathematical Sciences, Kyoto University, Kyoto-shi 606-8502, Japan
Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3469-3536
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The E6 state sum invariant is a topological invariant of closed 3–manifolds constructed by using the 6j–symbols of the E6 subfactor. In this paper, we introduce the E6 linear skein as a certain vector space motivated by E6 subfactor planar algebra, and develop its linear skein theory by showing many relations in it. By using this linear skein, we give an elementary self-contained construction of the E6 state sum invariant.

DOI : 10.2140/agt.2013.13.3469
Classification : 57M27, 57M15, 46L37
Keywords: state sum invariant, Turaev–Viro–Ocneanu invariant, $E_6$ subfactor planar algebra, $3$–manifolds, triangulation, linear skein

Okazaki, Kenta  1

1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto-shi 606-8502, Japan 
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Okazaki, Kenta. The state sum invariant of 3–manifolds constructed from the E6 linear skein. Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3469-3536. doi: 10.2140/agt.2013.13.3469

[1] S Bigelow, Skein theory for the ADE planar algebras, J. Pure Appl. Algebra 214 (2010) 658

[2] D E Evans, Y Kawahigashi, Quantum symmetries on operator algebras, Oxford University Press (1998)

[3] M Izumi, Subalgebras of infinite $C^{*}$–algebras with finite Watatani indices, I: Cuntz algebras, Comm. Math. Phys. 155 (1993) 157

[4] V Jones, Planar algebras, I

[5] L H Kauffman, S L Lins, Temperley–Lieb recoupling theory and invariants of $3$–manifolds, Annals Math. Studies 134, Princeton Univ. Press (1994)

[6] V Kodiyalam, V S Sunder, Topological quantum field theories from subfactors, Research Notes in Mathematics 423, Chapman Hall/CRC (2001)

[7] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997)

[8] S Morrison, E Peters, N Snyder, Skein theory for the $D_{2n}$ planar algebras, J. Pure Appl. Algebra 214 (2010) 117

[9] A Ocneanu, Chirality for operator algebras, from: "Subfactors" (editors H Araki, Y Kawahigashi, H Kosaki), World Sci. Publ. (1994) 39

[10] U Pachner, PL homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991) 129

[11] N Sato, M Wakui, Computations of Turaev–Viro–Ocneanu invariants of $3$–manifolds from subfactors, J. Knot Theory Ramifications 12 (2003) 543

[12] K Suzuki, M Wakui, On the Turaev–Viro–Ocneanu invariant of $3$–manifolds derived from the $E_6$–subfactor, Kyushu J. Math. 56 (2002) 59

[13] V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Mathematics 18, de Gruyter (1994)

[14] V G Turaev, O Y Viro, State sum invariants of $3$–manifolds and quantum $6j$–symbols, Topology 31 (1992) 865

[15] M Wakui, Turaev–Viro–Ocneanu invariants of $3$–manifolds constructed from quantum $6j$–symbols of the Coxeter graph $E_6$, Sūrikaisekikenkyūsho Kōkyūroku (1998) 6

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