We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

We prove new upper and lower bounds on the number of homotopy moves required to tighten a closed curve on a compact orientable surface (with or without boundary) as much as possible. First, we prove that Ω(n 2 ) moves are required in the worst case to tighten a contractible closed curve on a surface with non-positive Euler characteristic, where n is the number of self-intersection points. Results of Hass and Scott imply a matching O(n 2 ) upper bound for contractible curves on orientable surfaces. Second, we prove that any closed curve on any orientable surface can be tightened as much as possible using at most O(n 4 ) homotopy moves. Except for a few special cases, only naïve exponential upper bounds were previously known for this problem.

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ 1 and γ 2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ 1 and γ 2 where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems.We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fréchet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.1 Introduction This paper considers computational questions pertaining to homotopies: in broad terms, a homotopy between two curves in a topological space is a continuous deformation between these two curves. This can be formalized either in a continuous setting, where it constitutes one of the fundamental constructs of algebraic topology, but also in a more discrete one, where the input is a simplicial, or more generally cellular description of a topological space; this latter setting will be the focus of this article. While considerably more restrictive than the more traditional mathematical settings, this setting

We prove that the problem of deciding whether a 2-or 3-dimensional simplicial complex embeds into R 3 is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in R 3 like unknot or 3-sphere recognition which are in NP ∩ co-NP (assuming the generalized Riemann hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from lowdimensional topology, most importantly Dehn fillings on link complements.