Journal of Knot Theory and Its Ramifications

The Kauffman bracket skein modules, , have been calculated for A=±1 for all 3-manifolds M by relating them to the -character varieties. We extend this description to the case when A is a 4th root of 1 and M is either a surface×[0,1] or a rational homology sphere (or its submanifold).

Given a Bianchi Group , and a Hyperbolic manifold M, where π₁(M) is of finite index in Γ_d, we show that all boundary slopes are realized as the boundary slope of an immersed totally geodesic surface and hence are virtually embedded boundary slopes.

A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth.

We derive explicit formulas for the Reshetikhin–Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra in terms of the Seifert invariants and standard data for . A main corollary is a determination of the full asymptotic expansions of these invariants for lens spaces in the limit of large quantum level. This result is in agreement with the asymptotic expansion conjecture due to Andersen [1,2].

We introduce ribbon-moves of 2-links, which are operations to make 2-links into new 2-links by local operations.

Let L₁ and L₂ be 2-links. Then the following hold.

(1) If L₁ is ribbon-move equivalent to L₂, then we have

(2) Suppose that L₁ is ribbon-move equivalent to L₂. Let W_i be arbitrary Seifert hypersurfaces for L_i. Then the torsion part of H₁(W₁)⊕H₁(W₂) is isomorphic to G⊕G for a finite abelian group G.

(3) Not all 2-knots are ribbon-move equivalent to the trivial 2-knot.

(4) The converse of (1) is not true.

(5) The converse of (2) is not true.

Let L=(L₁,L) be a sublink of homology boundary link. Then we have:

(i) L is ribbon-move equivalent to a boundary link.

(ii) μ(L)=μ(L₁)+μ(L₂).

We introduce a new relation for high dimensional non-spherical knots, which is motivated by the codimension two surgery theory: a knot is a pull back of another knot if the former is obtained as the inverse image of the latter by a certain degree one map between the ambient spheres. We show that this relation defines a partial order for (2n-1)-dimensional simple fibered knots for n≥3. We also give some related results concerning cobordisms and isotopies of knots together with several important explicit examples.

We prove that the 3-sphere is a covering of itself branched over any volume zero knot.