## Representations of the Kauffman bracket skein algebra. II: Punctured surfaces.(English)Zbl 1422.57032

Summary: In Part I [the authors, Invent. Math. 204, No. 1, 195–243 (2016; Zbl 1383.57015)], we constructed invariants of irreducible finite-dimensional representations of the Kauffman bracket skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that is lifted in subsequent work relying on this one. A step in the proof is of independent interest, and describes the arithmetic structure of the Thurston intersection form on the space of integer weight systems for a train track.

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57R56 Topological quantum field theories (aspects of differential topology)

### Keywords:

Kauffman bracket; skein algebra; quantum Teichmüller space

Zbl 1383.57015
Full Text:

### References:

 [1] 10.1017/S0305004198003168 · Zbl 0939.57019 [2] 10.1016/0040-9383(94)00051-4 · Zbl 0887.57009 [3] 10.5802/afst.829 · Zbl 0880.57005 [4] 10.2140/gt.2007.11.889 · Zbl 1134.57008 [5] 10.1090/conm/560/11099 · Zbl 1333.57022 [6] 10.2140/gt.2011.15.1569 · Zbl 1227.57003 [7] 10.1090/proc/12927 · Zbl 1336.57043 [8] 10.1007/s00222-015-0611-y · Zbl 1383.57015 [9] 10.1090/S0002-9939-97-03943-9 · Zbl 0866.57005 [10] 10.1007/s000140050032 · Zbl 0907.57010 [11] 10.1142/S0218216599000183 · Zbl 0932.57015 [12] 10.1090/S0002-9939-02-06323-2 · Zbl 1059.57005 [13] 10.1090/S0002-9939-99-05043-1 · Zbl 0971.57021 [14] 10.1023/A:1022844520574 · Zbl 1087.81510 [15] ; Fok, Teoret. Mat. Fiz., 120, 511, (1999) [16] 10.1142/S0218216516500164 · Zbl 1378.57019 [17] 10.1063/1.1328078 · Zbl 1032.17022 [18] 10.1023/A:1007460128279 · Zbl 0897.57014 [19] 10.2140/agt.2015.15.1093 · Zbl 1315.57027 [20] 10.1142/S0218216509007129 · Zbl 1204.57033 [21] 10.1007/978-3-642-57916-5 · Zbl 0797.14004 [22] 10.1515/9781400882458 · Zbl 0765.57001 [23] 10.1016/S0040-9383(98)00062-7 · Zbl 0958.57011 [24] 10.1007/BF01239527 · Zbl 0725.57007 [25] 10.24033/asens.1639 · Zbl 0758.57011 [26] ; Turaev, Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Mathematics, 18, (1994) · Zbl 0812.57003 [27] ; Witten, Comm. Math. Phys., 121, 351, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.