Journal of Knot Theory and Its Ramifications

We discuss the ribbon-move for 2-knots, which is a local move. Let K and K' be 2-knots. Then we have the following: Suppose that K and K' are ribbon-move equivalent.

(1) Let (respectively, ) be the ℤ-torsion submodule of the Alexander module (respectively, ). Then is isomorphic to not only as ℤ-modules but also as ℤ[t, t^-1]-modules.

(2) The Farber–Levine pairing for K is equivalent to that for K'.

(3) The set of the values of the ℚ/ℤ-valued invariants for K is equivalent to that for K'.

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and ℙ²-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3-manifolds to be formed.

In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in ℝ^d, d ≥ 3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interpret the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d - 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings, we introduce new homological operations on Hochschild complexes. We show in future work that these operations are in fact the Dyer–Lashof–Cohen operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the -invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t^-1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S³. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S³. This generalizes a result of Hoste.

In [2], Cantwell–Conlon introduced a knot invariant, called depth. In this paper, we discuss the depth of a foliation ℱ on a 3-manifold with ℱ being (*) "codimension one, transversely oriented, taut, and proper". We show that for each n > 0, there exists a 3-manifold M such that (minimal depth of ℱ on M with ℱ being (*) and having exactly one depth 0 leaf) is greater than (minimal depth of ℱ on M with ℱ being (*)) + n.