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.

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).