Alexander Polytopes

Here I want to document my experience and process using Snappy to compute Alexander polytopes and Thurston Polytopes.

Let $\Delta_L$ denote the Alexander polynomial for a link $L$. In order to compute the Polytopes we will need the following libraries and packages.

            
                import snappy                
            
                from sage.geometry.newton_polygon import NewtonPolygon
                from sage.geometry.polyhedron.plot import Projection

The code itself is relatively straightforward. Let us define the following functions which will compute the Alexander polytopes.

            
                def Alex_Polytope_Unnormalized(L):
                    L_alex = L.alexander_polynomial()
                    if type(L_alex) == int:
                        return "Constant"
                    return Polyhedron(vertices = L_alex.exponents())
            
            
                def Alex_Polytope_True(L):
                    q = Alex_Polytope_Unnormalized(L)
                    if q == "Constant":
                        return "Constant"
                    q.center()
                    qnew = q - q.center()
                    qnewp = qnew.polar()
                    return (1/2)*qnewp
                
                def Alex_Polytope_True_Plot(L):
                    return Alex_Polytope_True(L).plot(frame=False, alpha=.6)

We can also compute the slopes for our Alexander polytopes.

            
                def Slope(L):
                    P = Alex_Polytope_True(L)
                    if P == "Constant":
                        return "constant"
                    if len(list(P.faces(4))) != 0:
                        return 
                    v = len(list(P.faces(0)))
                    f = len(list(P.faces(2)))
                    return (f-4)/(v-4)

Now let us look at a couple of examples.