The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge...

By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of nonconcordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has nontrivial Alexander polynomial, then there exists an infinite family of non-conc...

PURPOSE To assess the effect of cyanoacrylate glue on knot slippage and strength in vitro. METHODS Ethibond 3/0 was tested using a tensile tester. A single drop of cyanoacrylate glue was applied to the knots. Seven knot configurations (10 samples each) were tested: (1) a surgeon's knot (S) plus 4 square throws (+4) without glue (control group), (2) S plus one square throw (+1) without glue, (...

The cystine knot three-dimensional structure is found in many extracellular molecules and is conserved among divergent species. The identification of proteins with a cystine knot structure is difficult by commonly used pairwise alignments because the sequence homology among these proteins is low. Taking advantage of complete genome sequences in diverse organisms, we used a complementary approac...

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...

A 15-month-old girl with hydrocephalus from an intraventricular hemorrhage presented to the emergency department with irritability, lethargy and vomiting. Our neurosurgery team had inserted a ventriculoperitoneal shunt 10 months earlier. Computed tomography of the patient’s brain showed enlarged ventricles relative to postoperative imaging from when she had been asymptomatic. The child was brou...