Abstract
The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Grobner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Grobner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.
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