Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, it allows one to answer questions like Is there a direct relation between Khovanov homology and the A-polynomial of a knot? which would not have been asked otherwis...

The conormal Lagrangian LK of a knotK in R3 is the submanifold of the cotangent bundle T ∗R3 consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S∗R3, one obtains the unit conormal ΛK , and the Legendrian contact homology of ΛK is a knot invariant of K, known as knot contact homology. We define a version of string topology for str...

Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots and observe several new and unexpected phenomena. In the classical setting, if the longitude of a knot is trivial in the knot group then the group is infinite ...

Simple knots with just a few crossings can be made into attractive 3D geometric sculptures. Various approaches to achieve such a transformation are discussed and illustrated with examples. High-lighted are two larger-scale sculptures based on knots which were assembled in a few hours by participants of G4G9. 1. Knot Theory Primer To a mathematician, knots are closed curves embedded in three-dim...

Suppose that a hyperbolic knot in S admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or halfintegral, and they conjectured that the latter case does not happen. Using the correction terms in Heegaard Floer homology, we prove that if a hyperbolic knot in S admits a half-integral finite surgery, then the knot must have the same knot Floer homology as ...

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n > 1 the (n + 1)st iterated Bing double of a knot is rationally slice if and only if the nth iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying a given link. We prove certain similar geometric obstructions for n ≤ 1 as ...

The accidental untying of a shoelace while walking often occurs without warning. In this paper, we discuss the series of events that lead to a shoelace knot becoming untied. First, the repeated impact of the shoe on the floor during walking serves to loosen the knot. Then, the whipping motions of the free ends of the laces caused by the leg swing produce slipping of the laces. This leads to eve...

We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n–fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for thi...

This article pursues the study of the knot state asymptotics in the large level limit initiated in [CM11]. As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU2character manifold of the peripheral torus. ...