Journal of Knot Theory and Its Ramifications

We give some relationships of the Jones and Q polynomials between two links which are related by a band surgery. Then we consider two applications: The first one is to an evaluation of the ribbon-fusion number, the least fusion number of a ribbon knot. The second one is to DNA knot theory, helping us to understand the action of the Xer site-specific recombination at psi site.

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on ℙ²-irreducible manifolds. Moreover, for ℙ²-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S³, the projective space ℝℙ³ and the lens space L_4,1, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity.

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots.

Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form given a recursion relation for and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings.

The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

Let be the space of long j-knots in ℝⁿ. In this paper, we introduce a graph complex and a linear map via configuration space integral, and prove that (1) when both n > j ≥ 3 are odd, I is a cochain map if restricted to graphs with at most one loop component, (2) when n - j ≥ 2 is even, I is a cochain map if restricted to tree graphs, and (3) when n - j ≥ 3 is odd, I added a correction term produces a (2n - 3j - 3)-cocycle of which gives a new formulation of the Haefliger invariant when n = 6k, j = 4k - 1 for some k.

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link could give the same three-manifold M.

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.

We establish the volume conjecture for (m, 2)-cables of the figure 8 knot, when m is odd. For (m, 2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the figure 8 knot is false if one considers all the colors, but holds true if one restricts the colors to a subset of the set of positive integers.

The algebraic concordance group contains elements of order two, four, and of infinite order. Elements of infinite order are detected by the signature function. This paper develops computable invariants to simplify the computation of the order of torsion classes. The results are applied to determine the algebraic orders of all prime knots of 12 or fewer crossings.