Abstract
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n, n) Springer variety. The results build on Khovanov’s work with crossingless matchings and the cohomology of the (n, n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.
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References
Bar-Natan D.: Khovanov homology for tangles and cobordisms. Geom. Topol. 9, 1143–1199 (2005)
Asaeda M., Frohman C.: A note on the Bar-Natan skein module. Int. J. Math. 18, 1225–1243 (2007) math.GT/0602262v1
Kaiser, U.: Frobenius algebras and skein modules of surfaces in 3-manifolds. Algebraic Topology: Old and New, p. 85. Banach Center Publications (to appear) (2009). math.GT/0802.4068v1
Khovanov M.: Crossingless matchings and the (n,n) Springer variety. Commun. Contemp. Math. 6(2), 561–577 (2004) math.QA/0103190
Khovanov M.: A functor-valued invariant of tangles. Algebra Geom. Topol. 2, 665–741 (2002) math.QA/0103190
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Russell, H.M. The Bar-Natan skein module of the solid torus and the homology of (n, n) Springer varieties. Geom Dedicata 142, 71–89 (2009). https://doi.org/10.1007/s10711-009-9359-0
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DOI: https://doi.org/10.1007/s10711-009-9359-0