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