Knot theory is not generally considered an algebraic subject, due to the fact that knots don’t have much algebraic structure: there are a few operations defined on them (such as connected sum and cabling), but these don’t nearly make the space of knots finitely generated. In this thesis, following an idea of Dror Bar-Natan’s, we develop an algebraic setting for knot theory by considering the larger, richer space of knotted trivalent graphs (KTGs), which includes knots and links. KTGs along with standard operations defined on them form a finitely generated algebraic structure, in which many topological knot properties are definable using simple formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way to study knots. We present a construction for such an invariant. The starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90’s using the theory of associators. We present an elementary construction building on Kontsevich’s original definition, and discuss the homomorphicity properties of the resulting invariant, which turns out to be homomorphic with respect to almost all of the KTG operations except for one, called “edge unzip”. Unfortunately, edge unzip is crucial for finite generation, and we prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once. To fix this, we present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains ii all the good properties of the original KTGs: it is finitely generated, includes knots, and is closely related to Drinfel’d associators. The thesis is based on two articles, one published [Da] and one preprint [BD1], the second one joint with Dror Bar-Natan.

Knot theory is not generally considered an algebraic subject, due to the fact that knots don’t have much algebraic structure: there are a few operations defined on them (such as connected sum and cabling), but these don’t nearly make the space of knots finitely generated. In this thesis, following an idea of Dror Bar-Natan’s, we develop an algebraic setting for knot theory by considering the larger, richer space of knotted trivalent graphs (KTGs), which includes knots and links. KTGs along with standard operations defined on them form a finitely generated algebraic structure, in which many topological knot properties are definable using simple formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way to study knots. We present a construction for such an invariant. The starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90’s using the theory of associators. We present an elementary construction building on Kontsevich’s original definition, and discuss the homomorphicity properties of the resulting invariant, which turns out to be homomorphic with respect to almost all of the KTG operations except for one, called “edge unzip”. Unfortunately, edge unzip is crucial for finite generation, and we prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once. To fix this, we present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains ii all the good properties of the original KTGs: it is finitely generated, includes knots, and is closely related to Drinfel’d associators. The thesis is based on two articles, one published [Da] and one preprint [BD1], the second one joint with Dror Bar-Natan.

Doctor of Philosophy / Department of Mathematics / David R. Auckly / We study geometric variational problems for a class of models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. The Hopfions exhibit distinct localized knot-like structure and received a lot of attention lately in both mathematical and physical literature. High non-linearity of the energy functional presents both analytical and algebraic difficulties for studying it. In particular we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homotopy for smooth and continuous maps. As the spaces in question are neither linear nor even convex we take advantage of the algebraic structure on homogeneous spaces to represent maps by gauge potentials that form a linear space and reformulate the problem in terms of these potentials. However this representation of maps introduces some gauge ambiguity into the picture and we work out 'gauge calculus' for the principal bundles involved to apply the gauge-fixing techniques that eliminate the ambiguity. These bundles arise as pullbacks of the structure bundles H[arrow pointing right with hook on tail]G[arrow pointing right]G/H of homogeneous spaces and we study their topology and geometry that are of independent interest. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers only in each 2-homotopy class.

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