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...

In mathematical knot theory, knots are defined as a smooth embedding of topological circle into R3 and all its images [1, 3, 4, 6, 7]. These knots have no open ends, such that they cannot be tied or untied. If a knot can deform (an ambient isopoty) into a geometric circle, then it belongs to a special subgroup of knot, called unknot. For a mathematical knot, if we cut at finite locations, then ...

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader’s theorem which states that a knot is posi...

In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the “original knot” fibers over the circle then the wild knot Λ also fibers over the circle. As a consequence, the universal covering of S − Λ is R. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.