Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps… Expand

We investigate the computational complexity of some problems in three-dimensional
topology and geometry. We show that the problem of determining a bound on the genus of a
knot in a 3-manifold, is… Expand

The problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted is considered and it is shown that this problem, UNKNOTTING PROBLEM, is in NP.Expand

AbstractThe authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.
Theorem 1.Let f be a general position arc or loop on an orientable… Expand

There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is… Expand

1—> M representing a. is either an embedding or a double cover of a one-sided embedded curve K. In the second case, C bounds a Moebius band in M and K is isotopic to the centre of this band.

This paper presents a new and unified approach to the existence theorems for least area surfaces in 3-manifolds. Introduction. A surface F smoothly embedded or immersed in a Riemannian manifold M is… Expand

We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some… Expand