Related papers: A knot bounding a grope of class n is n/2-trivial

A knot k is called ``strongly (n-1)-trivial.'' if there exists a projection of k, such that one can choose n crossings of the projection with the property that making the crossing changes corresponding to any of the $2^{n}-1$ nontrivial…

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a…

We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro…

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…

A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds,…

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K)…

We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in…

Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not…

In this paper, the easier methods of my thesis are applied to give a simple proof of a theorem of Goussarov. The theorem relates two possible notions of finite type equivalence of knots, links or string links, showing that the resulting…

We prove the nontriviality, at all integral levels n, of the filtration, F_n, of the classical topological knot concordance group recently defined by the authors and Kent Orr [COT]. Recall that this filtration is significant not only…

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half…

Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and…

Let n be a positive integer. We provide a Khovanov homology proof of the following classical fact: If the closure of an n-strand braid is the n-component unlink, then the braid is trivial.

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology. We show that if a knot $K$ has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all…

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…

In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot…

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the…

Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by…