Journal of Knot Theory and Its Ramifications

A knot K in a closed connected orientable 3-manifold M is called a double-torus knot, if it is in a genus two Heegaard splitting surface H of M. We give a necessary and sufficient condition for a double-torus knot to be the trivial knot in words of meridian disks of genus two handlebodies obtained by splitting M along H.

Weight systems are constructed with solvable Lie algebras and their infinite dimensional representations. With a Heisenberg Lie algebra and its polynomial representations, the derived weight system vanishes on Jacobi diagrams with positive loop-degree on a circle, and it is proved that the derived knot invariant is the inverse of the Alexander-Conway polynomial.

Pulling back the weight system associated with the spinor representation of the Lie algebra by the universal Vassiliev-Kontsevich invariant yields a numerical link invariant with values in formal power series. Computing some skein relations satisfied by this invariant, I derive a recursive algorithm for its evaluation. The values of this invariant belong to the ring ℤ[W,W^-1] of Laurent polynomials in one variable.

Alexander polynomials of sextics are computed in the case of sextics with only simple singularities or sextics of torus type with arbitrary singularities. We will show that for irreducible sextics, there are only 4 possible Alexander polynomials: (t²-t+1)^j, j=0,1,2,3. For the computation, we use the method of Libgober and Loeser-Vaquié [5, 7] and the classification result in our previous papers [12, 11].

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in ℝ³ is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links.

We define invariants for templates that appear in certain dynamical systems. Invariants are derived from certain bialgebras. Diagrammatic relations between projections of templates and the algebraic structures are used to define invariants. We also construct 3-manifolds via framed links associated to tamplate diagrams, so that any 3-manifold invariant can be used as a template invariant.

We study trivalent graphs in S³ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a level edge connecting two vertices.

For any given knot K, a thick realization K₀ of K is a knot of unit thickness which is of the same knot type as K. In this paper, we show that there exists a family of prime knots {K_n} with the property that Cr(K_n)→∞(as n→∞) such that the arc-length of any thick realization of K_n will grow at least linearly with respect to Cr(K_n).

Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.