Journal of Knot Theory and Its Ramifications

We develop a skein recoupling theory for quantum G₂ involving skeins of 1 or 2 strands. The recoupling theory is used to define a combinatorial method for computing quantum G₂ 3-manifold invariants at 15th and 18th roots of unity.

Let K be a prime knot in S³ and G(K) = π₁(S³ - K) the knot group. We write K₁ ≥ K₂ if there exists a surjective homomorphism from G(K₁) onto G(K₂). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

A projection of the bouquet graph B with two cycles is said to be trivial if only trivial embeddings are obtained from the projection. In this paper a finite set of nontrivial embeddings of B is shown to be minimal among those which produce all nontrivial projections of B.

We give a simple argument to show that every polynomial f(t) ∈ ℤ[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links.

We consider a surface embedded in the 3-space. Such a surface is called unknotted if the closures of the complementary regions in the 3-sphere are both handlebodies. In this paper, we give a sufficient condition for a torus in the 3-space to be unknotted by using a projection onto a plane. We also prove that this condition is not sufficient for surfaces of higher genera.