Problems in Low Dimensional Topology

DMS-0306062 William Menasco The investigator's research is focussed on understanding two important phenomena in low dimensional topology: first, understanding exactly what is accomplished through the use of stabilization in relating two equivalent closed braids; and second, understanding when there is the occurence of topologically essential surfaces inside a 3-dimensional manifold, i.e. when is a 3-manifold Haken. The classical stabilization result is "Markov's Theorem" which says that any two closed braid representatives of the same oriented link type in the 3-sphere are related to each other through a sequence of moves (isotopies): conjugation, stabilization and destabilization. Unforunately, the Markov Theorem only says a sequence exists, but understanding exactly what this stabilization sequence accomplishes has largely remained a big black box until recently. The first peek inside the stabilization black box is the "Markov Theorem Without Stabilization" (MTWS), a product of a long collaborative effort with Joan Birman. The MTWS can tell one exactly what stabilization achieves in an isotopy between braids and one of the main goals of the project is to exploit this understanding in the areas of link classification, link invariants and contact geometry. An essential surface inside a 3-manifold tells one about the geometry of the space. Not all 3-manifolds contain essential surfaces, but it is possible that for a given 3-manifold M that is lacking any essential surface there is another 3-manifold M' which has essential surfaces and M' "covers" M. Understanding when a 3-manifold M has a such a corresponding cover M' is the focus of the "Virtual Haken Conjecture". The investigator in collaboration with Joseph Masters & Xingru Zhang is pursuing a new strategy for attacking this conjecture in a general setting. The 3-dimensional space in which we live is unique in its ability to retain information about the "knottest" of a collection of closed loops (think of a collection of tangled strands of pearls inside a jewelry box). This phemomenon of knottest does not occur in any lower or higher dimensional space---the advantage of a 5-dimensional jewelry box is that a collection of strands of pearls can never be tangled. Thus, as any micro-biologist working with DNA strands will tell you, knottest is an important feature of our 3-dimensional existence that needs to be understood. Basic questions arise. When are different two knots (single strands) or links (multiple strands) illustrating the same type of knottest---that is, when is there a sequence of motions of one link that move it around in 3-space so that it appears like that other link? When is When is a knot which appears to be tangle in fact equivalent through motions a simple circle that can be laid flat in a plane? Motions can be very complex (think of trying to untangle a mass of fishing line). The investigator's research has focussed on understanding and codifying these motions (in collaboration with Joan Birman, "The Markov Theorem Without Stabilization"). Also in trying to understand 3-dimensional spaces we can study "essential" surfaces that occur in them, i.e. surfaces that in the space can not be crashed down to a point. Such surfaces can give us a type of coordinate system for navigating in the space---astronomers are very interested in determining if our universe has any essential surfaces. If a 3-dimensional space has an essential surface then it is called "Haken". Not all spaces are Haken, but some that are not can be 'painted over' or "covered" by ones that are, i.e. they may be "Virtually Haken". The investigator in collaboration with Joseph Masters & Xingru Zhang is pursuing a new strategy determining when a 3-dimensional space is Virtually Haken.