When is a graph knotted?

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Bernstein, Gilbert

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Knot theory, as traditionally studied, asks whether or not a loop of string is knotted. That is, can we deform the loop in question into a circle without cutting or breaking it. In this thesis, I take a less traditional approach, studying networks of points connected together by string (i.e. a graph) instead of loops. By tracing different paths through this network we can identify many loops (i.e. cycles) in the network, each of which may or may not be knotted. Perhaps surprisingly, there will always be some knotted loop in a sufficiently complicated network. Such “sufficiently complicated” networks are called intrinsically knotted graphs. Very complicated graphs are always intrinsically knotted, and very simple graphs are always not, but graphs in between may be harder to identify. In this thesis, I present a method to reduce the question “Is the graph G intrinsically knotted?” to a linear algebra problem mod 2. Using this method I present a computer program that systematizes intrinsic knotting proofs and subsumes previous proof techniques. This program may lead to a conjecture for the intrinsic knotting obstruction set.



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