Alexander method

If an Alexander system fills a surface, a homeomorphism preserving every member up to isotopy is determined up to isotopy by its induced automorphism of the structure graph. A trivial graph action makes it isotopic to the identity; the finite graph automorphism group shows that the pointwise curve-class stabilizer is finite.

A homeomorphism of a closed disc that fixes its boundary pointwise is isotopic relative to the boundary to the identity. Radially interpolate the action toward the center after identifying the disc with the unit ball.

The center of $\mathrm{Mod}(T^2)$ has order two. A central class commutes with twists about two curves intersecting once, hence preserves both curve classes. The Alexander method reduces it to the identity or the elliptic involution $\iota :x\mapsto -x$, and both are central.