Journal of Knot Theory and Its Ramifications

The question whether or not a Dehn surgery on a knot in a lens space yields a lens space of the same order is investigated with homological techniques. Determining the first homology group of the lens space after surgery and of its covering yields some necessary conditions on the knot and the surgery curve. Application of these results along with a calculation of Seifert invariants answers the question completely for surgery on torus knots along nullhomological curves.

It is well-known that every 3-manifold M³ may be represented by a framed link (L,c), which indicates the Dehn-surgery from to M³ = M³(L,c); moreover, M³ is the boundary of a PL 4-manifold M⁴ = M⁴(L, c), which is obtained from by adding 2-handles along the framed link (L, c).

In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing M⁴=M⁴(L,c), directly from a planar diagram of (L,c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of M³=M³(L,c) and the regular genus of M⁴=M⁴(L,c) are obtained.

Let Z(M) be the 3-manifold invariant of Le, Murakami and Ohtsuki. We give a direct computational proof that the degree 1 part of Z(M) satisfies , where b₁(M) denotes the first Betti number of M and where λ_M denotes the Lescop generalization of the Casson-Walker invariant of M. Moreover, if b₁(M)=2, we show that Z(M) is determined by λ_M.

Invariants of virtual 2n-tangles t are defined using an analog of the Temperley-Lieb algebra. The invariants yield information about the bracket polynomial of any link in which t embeds.

In this paper we consider the generalized Smith conjecture of codimension greater than two, which says that no periodic tranformation of S^l can have the tame knotted S^h as fixed point set if l-h>2 and h>3. Using the Brieskorn spheres, this paper gives the explicit counterexamples to show that the conjecture is false in the DIFF category for the following cases: (i) l-h is even more than 2 and l is odd; (ii) 2l≤3(h+1) and h+1≡0 (mod 4).

We introduce the self-linking number of a smooth closed curve α:S¹→ℝⁿ with respect to a 3-dimensional vector bundle over the curve, provided that some regularity conditions are satisfied. When n=3, this construction gives the classical self-linking number of a closed embedded curve with non-vanishing curvature [5]. We also look at some interesting particular cases, which correspond to the osculating or the orthogonal vector bundle of the curve.

An uncountable collection of arcs in S³ is constructed, each member of which is wild precisely at its endpoints, such that the fundamental groups of their complements are non-trivial, pairwise non-isomorphic, and indecomposable with respect to free products. The fundamental group of the complement of a certain Fox-Artin arc is also shown to be indecomposable.

Extending and reproving a recent result of D. Krebes, we give obstructions to the embedding of a tangle in a link.

Any ribbon torus-knot in 4-space is naturally associated with a virtual knot. We investigate a relationship between ribbon torus-knots and virtual knots from this viewpoint. We also give a new example of a non-classical virtual knots which can not be detected by the group and the Z-polynomial.

We give an example of an irreducible homology sphere with an involution acting non-trivally on Floer homology.