We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the $19$-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in $3$-manifold topology.

We present three"hard"diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small"standard"example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $\mathbb{S}^2$.

We show that for every n there are two simple curves on the torus intersecting at least n times without the two curves folding or spiraling with respect to each other. On the other hand, two simple curves in a punctured plane that intersect at least n times (and do not create any empty bigons) must either form a spiral of depth d or a fold of width cn/(d+1)−1, where c only depends on the number of punctures in the plane. The construction of the two curves on the torus involves train tracks and word equations, and the verification that the two curves do not spiral leads us to an infinite binary word based on the golden ratio which does not contain any square word ww for which |w| is even.

We prove that the problem of deciding whether a two- or three-dimensional simplicial complex embeds into R 3 is NP -hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an S 3 filling is NP -hard. The former stands in contrast with the lower-dimensional cases, which can be solved in linear time, and the latter with a variety of computational problems in 3-manifold topology, for example, unknot or 3-sphere recognition, which are in NP ∩ co- NP. (Membership of the latter problem in co-NP assumes the Generalized Riemann Hypotheses.) Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings of manifolds with boundary tori.

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.

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). 2012 ACM Subject Classification Mathematics of computing → Geometric topology; Theory of computation → Problems, reductions and completeness

We prove that deciding if a diagram of the unknot can be untangled using at most k Riedemeister 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 show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari\~no.

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and 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 low-dimensional topology, most importantly Dehn fillings on link complements.

We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R-3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S-3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S-3, then there is also an embedding in which X has a short meridian, that is, an essential curve in the boundary of X bounding a disk in S-3 \ X with length bounded by a computable function of the number of tetrahedra of X.

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We show here that the net twisting of the helical pieces of any such surface in a given triangulated 3-manifold is bounded.

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(n2) 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 0(n2) 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 0(n4) homotopy moves. Except for a few special cases, only naive exponential upper bounds were previously known for this problem.

We show that Heegaard Genus ≤ g, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to g, is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem CNF-SAT to Heegaard Genus ≤ g.

Let \(\mathcal{{T}}\) be a triangulation of a set \(\mathcal{{P}}\) of n points in the plane, and let e be an edge shared by two triangles in \(\mathcal{{T}}\) such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of \(\mathcal{{P}}\) from \(\mathcal{{T}}\). The flip distance between two triangulations of \(\mathcal{{P}}\) is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of \(\mathcal{{P}}\) is at most k, for some given \(k \in \mathbb {N}\). It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time \(\mathcal {O}(n + k \cdot c^{k})\), for a constant \(c \le 2 \cdot 14^{11}\), which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.

Let M be a 3-manifold with torus boundary components T_1 and T_2 . Let ϕ :T_1→ T_2 be a homeomorphism, M_ϕ the manifold obtained from M by gluing T_1 to T_2 via the map ϕ , and T the image of T_1 in M_ϕ . We show that if ϕ is “sufficiently complicated” then any incompressible or strongly irreducible surface in M_ϕ can be isotoped to be disjoint from T . It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a constrained way.

This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together researchers working on various aspects of triangulations and of fostering a closer collaboration within the computational geometry and topology community. Triangulations are highly suitable for computations due to their clear combinatorial structure. As a consequence, they have been successfully employed in discrete algorithms to solve purely theoretical problems in a broad variety of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold topology, geometry, and others). However, due to the large variety of applications, requirements vary from field to field and thus different types of triangulations, different tools, and different frameworks are used in different areas of research. This is why today closely related research areas are sometimes largely disjoint leaving potential reciprocal benefits unused. To address these potentials a workshop on Triangulations was held at Oberwolfach Research Institute in 2012. Since then many new collaborations between researchers of different mathematical communities have been established. Regarding the computational geometry community, the theory of manifolds continues to contribute to advances in more applied areas of the field. Many researchers are interested in fundamental mathematical research about triangulations and thus will benefit from a broad set of knowledge about different research areas using different techniques. We hope that this workshop brought together researchers from many different fields of computational geometry to have fruitful discussions which will lead to new interdisciplinary collaborations and solutions.

We show that a strongly irreducible and boundary-strongly irreducible surface can be isotoped to be almost normal in a triangulated 3-manifold.

We consider two systems ( α 1 , …, α m ) and ( β 1 , …, β n ) of simple curves drawn on a compact two-dimensional surface M with boundary. Each α i and each β j is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The α i are pairwise disjoint except for possibly sharing endpoints, and similarly for the β j . We want to “untangle” the β j from the ai by a self-homeomorphism of M ; more precisely, we seek a homeomorphism φ: M → M fixing the boundary of M pointwise such that the total number of crossings of the ai with the φ( β j ) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if M is planar, i.e., a sphere with h ≥ 0 boundary components (“holes”), then O ( mn ) crossings can be achieved (independently of h ), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable or nonorientable) genus g ≥ 0, we obtain an O(( m + n ) 4 ) upper bound, again independent of h and g . The proofs rely, among other things, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.

We show that for every n there are two simple curves on the torus intersecting at least n times without the two curves folding or spiralling with respect to each other. On the other hand, two simple curves in a punctured plane that intersect at least n times (and do not create any empty bigons) must either form a spiral of depth d or a fold of width cn/(d + 1) − 1, where c only depends on the number of punctures in the plane. The construction of the two curves on the torus involves train tracks and word equations, and the verification that the two curves do not spiral leads us to an infinite binary word based on the golden ratio which does not contain any square word ww for which |w| is even.