For null-homologous knots in rational homology 3-spheres, there are two equivariant invariants obtained by universal constructions à la Kontsevich, one due to Kricker and defined as a lift of the Kontsevich integral, and the other constructed by Lescop by means of integrals in configuration spaces. In order to explicit their universality properties and to compare them, we study a theory of finite type invariants of null-homologous knots in rational homology 3-spheres. We give a partial combinatorial description of the space of finite type invariants, graded by the degree. This description is complete for knots with a trivial Alexander polynomial, providing explicit universality properties for the Kricker lift and the Lescop equivariant invariant and proving the equivalence of these two invariants for such knots.[-]

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. As an application, if $\{K_n\}$ contains infinitely many L-space knots, then the latter must occur. We further develop this to show that if $K_n$ is an L-space knot for infinitely many integers $n > 0$ and infinitely many integers $n < 0$, then $c$ is a braid axis. We then use this to show that satellite L-space knots are braided satellites.
This is joint work with Ken Baker.[-]

A surface system for a link in $S^3$ is a collection of embedded Seifert surfaces for the components, that are allowed to intersect one another. When do two $n$–component links with the same pairwise linking numbers admit homeomorphic surface systems? It turns out this holds if and only if the link exteriors are bordant over the free abelian group $\mathbb{Z}_n$. In this talk we characterise these geometric conditions in terms of algebraic link invariants in two ways: first the triple linking numbers and then the fundamental groups of the links. This involves a detailed study of the indeterminacy of Milnor's triple linking numbers.
Based on joint work with Chris Davis, Matthias Nagel and Patrick Orson.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]