Hugh Howards

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

Thin position for knots and 3-manifolds: a unified approach

We unify the notions of thin position for knots and for 3-manifolds and survey recent work concerning these notions.

CONVEX BRUNNIAN LINKS

We prove that the Borromean Rings are the only Brunnian link of 3 or 4 components that can be built out of convex curves.

Intrinsic linking and knotting of graphs in arbitrary 3–manifolds

are intrinsically linked, and used these two results to provethat any graph with a minor in the Petersen family (Figure 1) is intrinsically linked.Conversely, Sachs conjectured that any graph which is intrinsically linked contains aminor in the Petersen family. In 1995, Robertson, Seymour and Thomas [10] provedSachs’ conjecture, and thus completely classified intrinsically linked graphs.Examples of intrinsically knotted graphs other than K

Limits of incompressible surfaces

Abstract One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater (Howards, 1998). This paper proves the contrasting, but not contradictory result that although one can sometimes embed arbitrarily many surfaces in a 3-manifold it is impossible to ever embed an infinite number of such surfaces in any compact, orientable 3-manifold M .

INTRINSIC LINKING IN DIRECTED GRAPHS

We extend the notion of intrinsic linking to directed graphs. We give methods of constructing intrinsically linked directed graphs, as well as complicated directed graphs that are not intrinsically linked. We prove that the double directed version of a graph G is intrinsically linked if and only if G is intrinsically linked. One Corollary is that J6, the complete symmetric directed graph on 6 vertices (with 30 directed edges), is intrinsically linked. We further extend this to show that it is possible to find a subgraph of J6 by deleting 6 edges that is still intrinsically linked, but that no subgraph of J6 obtained by deleting 7 edges is intrinsically linked. We also show that J6 with an arbitrary edge deleted is intrinsically linked, but if the wrong two edges are chosen, J6 with two edges deleted can be embedded linklessly.

FORMING THE BORROMEAN RINGS OUT OF ARBITRARY POLYGONAL UNKNOTS

We prove the perhaps surprising result that given any three polygonal unknots in

An infinite family of convex Brunnian links in {\mathbb{R}^n}

\R^3

Every graph has an embedding in

, then we may form the Borromean rings out of them through rigid motions of

S^3

\R^3