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Cross-ratio degrees count configurations of points z 1 , . . . , zn ∈ P 1 satisfying n − 3 cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be viewed as combinatorial invariants of certain hypergraphs. We prove an upper bound on cross-ratio degrees in terms of the theory of perfect matchings on bipartite graphs. We also discuss several of the many perspectives on cross-ratio degrees -including a connection to Gromov-Witten theory -and give many example computations.

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants,we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index. arXiv:1108.5700v1 [math.GT]