Triangular decomposition of skein algebras.(English)Zbl 1427.57011
The introduction of the Jones polynomial constituted a great breakthrough in knot theory. Big portion of the development in knot theory from that pivotal moment is devoted to deeper understanding of this discovery. One such very important and fruitful development was due to Kauffman who introduced the skein rules, which (almost) recreate the Jones polynomial by getting rules allowing for removing crossings with certain weights assigned to different possible resolutions of the crossings, and to the trivial knots that are eventually left by the process defined by the application of the first of the rules.
The Kauffman rules were used to introduce skein algebras byV. G. Turaev [Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635–704 (1991;Zbl 0758.57011)] andJ. H. Przytycki [Kobe J. Math. 16, No. 1, 45–66 (1999;Zbl 0947.57017)]. A generalization of the latter is the subject of the paper under review. The generalization concerns manifolds which are Cartesian products of surfaces that might have punctures and boundaries with a unit interval. Many of such surfaces have ideal triangulations and the paper concerns this case. It is proposed that the skein algebras are related to skein algebras associated to triangles, and the latter are related to Chekhov-Fock algebras that are deformed versions of Poisson algebras associated to triangles. This way there is a simple inclusion of the skein algebra associated to the surface into the tensor product of Chekhov-Fock algebras associated to ideal triangles of the triangulation. However, this requires some extra conditions on top of the standard skein relation and the trivial loop relation in order to have consistency on the glued edges of the triangles. As a (very important) byproduct, a formula for the quantum trace map that generalizes the formula ofF. Bonahon andH. Wong [Geom. Topol. 15, No. 3, 1569–1615 (2011;Zbl 1227.57003)] is found. Also, a relationship between the skein algebra byG. Muller [Quantum Topol. 7, No. 3, 435–503 (2016;Zbl 1375.13038)] and the one introduced in the paper is found.
The Kauffman rules were used to introduce skein algebras byV. G. Turaev [Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635–704 (1991;Zbl 0758.57011)] andJ. H. Przytycki [Kobe J. Math. 16, No. 1, 45–66 (1999;Zbl 0947.57017)]. A generalization of the latter is the subject of the paper under review. The generalization concerns manifolds which are Cartesian products of surfaces that might have punctures and boundaries with a unit interval. Many of such surfaces have ideal triangulations and the paper concerns this case. It is proposed that the skein algebras are related to skein algebras associated to triangles, and the latter are related to Chekhov-Fock algebras that are deformed versions of Poisson algebras associated to triangles. This way there is a simple inclusion of the skein algebra associated to the surface into the tensor product of Chekhov-Fock algebras associated to ideal triangles of the triangulation. However, this requires some extra conditions on top of the standard skein relation and the trivial loop relation in order to have consistency on the glued edges of the triangles. As a (very important) byproduct, a formula for the quantum trace map that generalizes the formula ofF. Bonahon andH. Wong [Geom. Topol. 15, No. 3, 1569–1615 (2011;Zbl 1227.57003)] is found. Also, a relationship between the skein algebra byG. Muller [Quantum Topol. 7, No. 3, 435–503 (2016;Zbl 1375.13038)] and the one introduced in the paper is found.
Reviewer: Robert Owczarek (Los Alamos)
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
