We show that the map on components from the space of classical long knots to the nth stage
of its Goodwillie–Weiss embedding calculus tower is a map of monoids whose target is an …

In this thesis we consider two homotopy theoretic approaches to the study of spaces of
knots: the theory of finite type invariants of Vassiliev and the embedding calculus of …

We associate a Taylor tower supplied by the calculus of the embedding functor to the space
of long knots and study its cohomology spectral sequence. The combinatorics of the spectral …

We show that the invariants $ ev_n $ of long knots in a $3 $-manifold, produced from
embedding calculus, are surjective for all $ n\geq1 $. On one hand, this solves some of the …

Using derived categories of equivariant coherent sheaves, we construct a categorification of
the tangle calculus associated to sl (2) and its standard representation. Our construction is …

We introduce explicit holonomy perturbations of the Chern–Simons functional on a 3–ball
containing a pair of unknotted arcs. These perturbations give us a concrete local method for …

The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot
(and consequently, all known abelian invariants), and converts knots to topologically slice …

In this paper, we show an isomorphism of homological knot invariants categorifying the
Reshetikhin–Turaev invariants for sl n. Over the past decade, such invariants have been …

We use a special kind of 2-dimensional extended Topological Quantum Field Theories
(TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's …

We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction
for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link …