Journal of Knot Theory and Its Ramifications

In 1986 Falk and Randell, based on Arnold's 1969 paper on braids, proved that the pure braid groups are residually nilpotent. They also proved that the quotients in the lower central series are free abelian groups.

This brief note uses an example to provide evidence for a much stronger conjectural statement: That each braid b can be written as an infinite sum , where each b_i is a linear function of the ith Vassiliev–Kontsevich Z_i(b) invariant of b. The example is pure braids on two strands. This leads to solving e^τ = q for τ a Laurent series in q. We set and use Fourier series and Parseval's theorem to prove e^τ = q.

For more than two strands the stronger statement seems to rely on an as yet unstated Plancherel theorem for braid groups, which is likely both to be deep and to have deep consequences.

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under five cube diagram moves. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules equipped with their Blanchfield forms ϕ, and the modules such that there is a unique isomorphism class of , and we prove that for the other modules , there are infinitely many such classes. We realize all these by explicit knots in ℚ-spheres.

We give an algorithm which decides the Nielsen–Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids.

We study a two-point self-avoidance energy which is defined for all rectifiable curves in ℝⁿ as the double integral along the curve of 1/r^q. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊¹ or to a closed interval I. For q > 2 the energy evaluated on curves in ℝ³ turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ³ with finite -energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q_crit = 2, the arclength parametrization of γ is of class C^{1, 1-2/q}, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids.

We give a description of Khovanov's knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1}ⁿ. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d² = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves.

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

In this paper, we compute the Khovanov homology over ℚ for (p, -p, q) pretzel knots for 3 ≤ p ≤ 15, p odd, and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p, -p, q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over ℚ or ℤ). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.

Enforcing a specific symmetry group on a curve, knotted or not, is not trivial using standard interpolations such as polygons or splines. For a prescribed symmetry group we present a symmetrization process based on a Fourier description of a knot. The presence of symmetry groups implies a characteristic pattern in the Fourier coefficients. The relations between the coefficients are shown for five ideal knot shapes with their proposed symmetry groups.