Electrical Reduction, Homotopy Moves, and Defect

Abstract

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, Ω(n 3/2 ) degree-1 reductions, series-parallel reductions, and ∆Y transformations are required in the worst case to reduce an n-vertex plane graph to a single vertex or edge. The lower bound is achieved by any planar graph with treewidth Θ( n). Second, Ω(n 3/2 ) homotopy moves are required in the worst case to reduce a closed curve in the plane with n self-intersection points to a simple closed curve. For both problems, the best upper bound known is O(n 2 ), and the only lower bound previously known was the trivial Ω(n).The first lower bound follows from the second using medial graph techniques ultimately due to Steinitz, together with more recent arguments of Noble and Welsh [J. Graph Theory 2000]. The lower bound on homotopy moves follows from an observation by Haiyashi et al. [J. Knot Theory Ramif. 2012] that the standard projections of certain torus knots have large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold.

Models of random knots

“…This means that c 2 drifts away from zero. Chang and Erickson [CE15] consider a generalization of the star model. They define the flat torus diagram T (p, q) as the closed braid (σ 1 σ 2 • • • σ p−1 ) q , and assign crossing signs at random.…”

Eigenvalue contour lines of Kac-Murdock-Szego matrices with a complex parameter

“…Evidently, our two rules are similar in nature to those used in computing winding numbers of closed, bounded, and oriented curves [8]. The latter rules, which appear in the literature in various contexts [10,11], are often referred to as Alexander numbering, the name originating from Alexander's 1928 paper [12]. More precisely, for the orientation provided in Fig.…”

Universal knot diagrams

“…(2) The star S n with 2n + 1 tips is a polygonal curve with (2n + 1) segments and (n − 1)(2n + 1) crossing points, as illustrated in Figure 2 for n = 2, 3, and 4. Star diagrams are special types of closed braids with n strands, and they coincide with (n, 2n + 1) torus knot projections [HHSY12,CE15]. See also [EHLN16].…”