We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in R3. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of K? The main result, Theorem 1 below,

When given a very tangled but unknotted circular piece of string it is usually quite easy to move it around and tug on parts of it until it untangles. However solving this problem by computer, either exactly or heuristically, is challenging. Various approaches have been tried, employing ideas from algebra, geometry, topology and optimization. This paper investigates the application of motion planning techniques to the untangling of mathematical knots. Such an approach brings together robotics and knotting at the intersection of these fields: rational manipulation of a physical model. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths for physical models. Using a probabilistic planner, we have untangled some standard benchmarks described by over four hundred variables much more quickly than has been achieved with minimization. We also show how to produce candidates with minimal number of segments for a given knot. We discuss novel motion planning techniques that were used in our algorithm and some possible applications of our untangling planner in computational topology and in the study of DNA rings.

Abstract. We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard. 1.

For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with

Given a closed polygon P having n edges, embedded in R d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P. We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7. Keywords: isoperimetric inequality, Plateau’s problem, computational complexity

We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r 2. In the direction of lower bounds, we give a sequence of length one curves with r→0for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A. 1

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

In the manuscript [2] the rst author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3-space. The boundaries of these `foliated embeddable discs' (FED's) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FED's. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list. In this paper we describe implementations of the rst and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FED's have unexplored and interesting structure. Open questions are interspersed throughout the manuscript. The third part of the algorithm was studied in [3] and [4], and implemented by S.J. Lee [20]. At this writing his algorithm is polynomial for n 4 and exponential for n 5. 1

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1