Re: This Week's Finds in Mathematical Physics (Week 228)


There's a simple topological interpretation of the element of the
rational projective line associated to a rational tangle. I don't know
how to use this to prove the theorem, and I don't know a reference for
it (maybe it is in one of the references you cited). Anyway, regard a
rational tangle as a two-component curve C in the 3-ball B^3 whose four
boundary points are on the 2-sphere S^2. Consider the double branched
cover of B^3 along C. This is a 3-manifold Y whose boundary can be
identified with the 2-torus T^2. (In fact Y is a solid torus.) The
inclusion of T^2 into Y induces a map from H_1(T^2) to H_1(Y), and the
kernel of this map is a one-dimensional subspace of H_1(T^2) = Z^2. If
I am not mistaken. this is the element in question of the rational
projective line.

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