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Invariants on Graphs Part I

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Dr Simrat Ghuman, SLU

  • Topology Seminar
When Fri, Oct 01, 2004
from 11:00 AM to 11:50 AM
Where RH 316
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Abstract: In this talk, we address a classical problem in low dimensional topology: the classification of tamely embedded, finite, connected graphs in the three-sphere up to ambient isotopy. In the case that the graph is homeomorphic to a circle, our problem reduces to the embedding problem for knots in the three-sphere. Our major result is the existence of a unique isotopy class of longitudes of a cycle for an infinite class of graphs. We then define new invariants for this infinite class of graphs. First we define a longitude of a cycle in the graph. In contrast to the situation of a knot, for a graph it is quite difficult to canonically select an isotopy class of longitudes. However we prove that longitudes exist for any cycle in any finite graph and are unique in certain situations. This definition of a longitude can be considered an extension of the definition of a longitude of a tamely embedded knot in the three-sphere. Next, in the situation in which the longitude for a cycle is unique, we define a sequence of invariants which detect whether this longitude lies in the n-th term of the lower central series of the fundamental group of the complement of the graph in the three-sphere. These invariants can be viewed as extensions of Milnor's invariants of a link. Although this invariant is not complete, we provide examples illustrating that this invariant is more sensitive than Milnor's invariant applied to a subgraph consisting of a link.

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