The set of maps F_{a,b}: x -> x+a+{b/{2 pi}} sin(2 pi x) with any given rotation interval is contractible

Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and adds references. v3 improves two figures and their captions. New version v4 offers a completely different proof of non-overlap in the quasigeodesic loop case, and contains several other substantive improvements. This version is 23 pages long, with 15 figure

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

Let $\gamma$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $\gamma$ is self-overlapping by geometrically constructing a combinatorial word from $\gamma$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $\gamma$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $\gamma$ with minimum area.Comment: 27 pages, 16 figure

Isometric endomorphisms of free groups

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.Comment: 26 pages, 6 figures; minor typographical edits for final published versio