Essential-surface complexes of two-bridge knots

The essential-surface complex $\ES(E)$ of a knot exterior $E$ has theisotopy classes of essential surfaces as vertices and collections ofpairwise disjoint surfaces as simplices; the symmetry group of theknot acts on it. We determine this complex and this action for alltwo-bridge knots, starting from the Hatcher--Thurston classification.The complex is the disjoint union of the subcomplexes carried by theindividual Hatcher--Thurston branched surfaces, so essential surfacescarried by different branched surfaces are never disjoint, and eachsubcomplex is computed by an explicit rule for the components of acarried surface. The two-bridge knots fall into three classes. Forthe torus knots $T(2,q)$ and the twist knots the complex consists oftwo or three isolated vertices, and these are exactly the two-bridgeknots with at most one essential surface for each boundary slope. Forthe knots with continued fraction $[a,b]$, $[a,1,b]$, or $[a,1,1,b]$with $a,b$ odd, the complex is a finite graph whose components arepaths with at most five vertices; these are exactly the remainingtwo-bridge knots with finitely many essential surfaces. For all othertwo-bridge knots the complex is an explicit infinite disjoint union offinite connected complexes, whose vertices are the lattice points ofpolytopes. We determine the kernel of the action for every hyperbolictwo-bridge knot: it is the whole orientation-preserving symmetry groupfor the twist knots, has order two for the other double twist knots,and is trivial otherwise. When the complex is decorated with surfacetypes and boundary slopes, the action is surjective onto theautomorphism group for the double twist knots, whereas for the knotswith infinitely many essential surfaces the automorphism group isuncountable. Along the way we determine the orientability and thenumber of components of every surface carried by a Hatcher--Thurstonbranched surface, describe the action of the standard symmetries onthe Hatcher--Thurston parameters, and correct the count of isotopyclasses of carried surfaces.