Low Dimensional Topology

September 8, 2008

Intrinsic linking/knotting in arbitrary manifolds

Filed under: 3-manifolds,Knot theory — Jesse Johnson @ 10:53 am

A graph is said to be intrinsically knotted/linked if for every embedding of the graph into the 3-sphere, there is a (cyclic) subgraph that forms a non-trivial knot/link.  If one wanted to generalize this to embeddings in an arbitrary 3-manifold, there are a couple of ways to proceed.

Flapan, Howards, Lawrence, and Mellor [1] gemeralized the notion as follows:  They call a graph intrinsically knotted/linked in a 3-manifold M if every embedding contains a (cyclic) subgraph that doesn’t form an unknot or have unlinked components, respectively, contained in a ball in the 3-manifold.  This definition gets the job done, but it doesn’t seem to take advantage of the extra topology in the arbitrary 3-manifold.  In fact, the authors show that a graph is intrinsically knotted/linked in an arbitrary 3-manifold if and only if it is intrinsically knotted/linked in the 3-sphere.

Another way to define it, proposed by Foisy, Bustamante, Federman,  Kozai, Matthews, McNamara, Stark and Trickey* [2] is as follows:  A graph is intrinsically knotted in a given 3-manifold M if there is no embedding into M in which each cyclic subgraph is isotopic to a geodesic or to an unknot contained in a ball.  (Actually, this definition does not appear to be in the paper cited, but was suggested by Joel Foisy in a talk I saw last year.)  A graph is intrinsically linked in M if every embedding contains a two-component link in which neither component is contained in a ball.

Note that it is possible for a graph to be intrinsically linked by the first definition but not by the second definition if there is an embedding in which some subgraph forms a link in which one but not both components are contained in a ball.  In fact, the authors of the second paper demonstrate an embedding of K_6 (the complete graph with six vertices) in projective 3-space with just this property.  The graph K_6 is long known to be intrinsically linked, but in the embedding they produce, every 2-component cyclic subgraph has a component contained in a ball that is disjoint from the other (possbly non-trivial) component.  (Though I wonder if there’s an even more general definition of unlinked in an arbirtrary manifold…)

Working with this second definition of intrinsically knotted graphs in an arbitrary 3-manifold is much more difficult because it’s harder to tell when a loop is isotopic to a geodesic than when it bounds an embedded disk.  I vaguely remember Joel mentioning results about intrinsically knotted graphs in RP^3 during his talk, but it’s not mentioned in the paper above.

(*You may have noticed that there are eight names on the second paper cited above.  That’s because it’s the result of a summer REU (Research Experience for Undergraduates) at SUNY Potsdam.  Intrinsic linking appears to be a good topic these projects; the Potsdam REU produced a paper [3] on intrinsic linking last year as well and Thomas Mattman’s REU at Cal. State Chico produced a paper [4] the year before that.)

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