In [23] we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of th...

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.

The knot vectors of a B-spline surface determine the basis functions hereby, together with the control points, the shape of the surface. Knot manipulations and their influence on the shape of curves have been investigated in several papers (see e.g. [4] and [5]). The computations can be made very efficiently, if the basis functions and the vector function of the B-spline surface are represented...

The Nematodes are roundworms that are found in every environment of the earth. While some species are harmful parasites, others play a vital role in nutrient cycle and medical research. Nematode infestation in the fields is poly-specific; however, depending on the agro-climatic conditions, one or two species are dominant over the rest. The present studies attempts to observe and control the roo...

We study satellites of Legendrian knots in R and their relation to the Chekanov–Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We...

During arthroscopic knot tying, one of the suture limbs may break before completion of the knot. With a single suture limb left, it may not be possible to secure the knot against slipping, and the surgeon may be forced to use an additional anchor to obtain a secure repair. We present a tool and method that allow completion of the knot on a single suture limb. Alternatively, the method can be us...

Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the s...

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0or 1-crossing knot. The proof uses techniques from sutured manifold theory.

Given a 3-manifold M , there are generically infinitely many manifold which covers M . However, if we are restricted to the category of knot complements, the situation is quite different. It can be shown (see Lemma 1 and bellow) that if the complement E(K) of a knot K is n-fold covered by some knot complement, then the covering is cyclic, and K admits a cyclic surgery, i.e. a Dehn surgery such ...

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (with possible...