Journal of Knot Theory and Its Ramifications

We prove in this paper that any 2 dimensional modular functor satisfying that S_1,1≠0 induces a family of 2+1 dimensionally topological quantum field theories. We do this for two kinds of modular functors namely a modular functor on the category of extended surfaces and a modular functor on the category of extended surfaces with marked points and directions. We follow the ideas of M. Kontsevich, [21], and K. Walker, [32] but we give proofs and provide details left out in [21] and [32]. Careful study also shows that more choices are needed to define the TQFT than it is revealed in [21] and [32]. On the other hand, relations found in [32] turns out not to be needed here.

In this paper, we use knots with bands ("kwb") to define a calculus for smoothly embedded surfaces in 4-space. This approach has its roots in the classical method of moving pictures of hyperplane slices, and thus shares its simplicity and naturality. Unlike moving pictures, however, kwb are combinatorial knotlike objects in 3-space (which can, in turn, be represented by planar diagrams), and a set of three simple combinatorial moves is sufficient to connect the kwb representing isotopic surfaces. Thus we obtain a one-to-one correspondence between isotopy classes of embedded surfaces and certain kwb modulo these moves, answering a question first posed in 1994 by Katsuyuki Yoshikawa.

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ³ contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ³ that contains no non-split n-component link.

We show how an incompressible Seifert surface F for a knot K in S³ can be used to create an essential lamination ℒ_F in the complement of each of an infinite class of knots associated to F. This lamination is persistent for these knots; it remains essential under all non-trivial Dehn fillings of the knot complement. This implies a very strong form of Property P for each of these knots.

By using Ghrist's theorem that universal templates exist, we define a family of integer-valued invariants {ν_n} of oriented knot types, each of which is complete. We give (1) an estimation of ν_n(K) from below in terms of the genus of K and (2) an estimation of ν_n(K) from above in terms of the word length of a braid presentation of K. To do this, we introduce a yet another invariant λ_n(K) and give an estimation of λ_n(K). Moreover, we show that the estimation of λ_n(K) from below is best possible, by explicitly determining the invariant ν_n(K) for some knots.

One of the most widely used methods to build yield curve models is to use principal components analysis on the correlation matrix of the innovations. R. Litterman and J. Scheinkman found that three factors are enough to explain most of the moves in the case of the US treasury curve. These factors are level, steepness and curvature. Working in the context of commodity futures, G. Cortazar and E. Schwartz found that the spectral structure of the correlation matrices is strikingly similar to those found by R. Litterman and J. Scheinkman. We observe that in both cases the correlation between two different contracts maturing at times t and s is roughly of the form ρ^|t-s|, for a certain (fixed) 0≤ρ≤1. Assuming this correlation structure we prove that the observed factors are perturbations of cosine waves.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω₃-moves that only differ in how the three strands that are involved in the move are ordered on the knot.

To transform knot diagrams of isotopic knots into each other one must in general use Ω₃-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω₃-moves are introduced.

Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

We show that if a split link is obtained from a split link L in S³ by 1/n-Dehn surgery along a trivial knot C, then the link L∪C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial link is obtained from a trivial link via a twisting.

We propose a transfer matrix algorithm for the enumeration of alternating link diagrams with external legs, giving a weight n to each connected component. Considering more general tetravalent diagrams with self-intersections and tangencies allows us to treat topological (flype) equivalences. This is done by means of a finite renormalization scheme for an associated matrix model. We give results, expressed as polynomials in n, for the various generating functions up to order 19 (2-legged link diagrams), 15 (tangles) and 11 (6-legged links) crossings. The limit n→∞ is solved explicitly. We then analyze the large-order asymptotics of the generating functions. For 0≤n≤2 good agreement is found with a conjecture for the critical exponent, based on the KPZ relation.

It is shown that if the defining word of a cyclically presented group is admissable then its natural HNN extension is the group of a high dimensional knot. As an example we define a family of cyclically presented groups which contains Sieradski groups, Fibonacci groups, and Gilbert-Howie groups. It is proven that HNN extensions of these groups are LOG groups and so, are fundamental groups of complements of codimension two closed orientable connected tamely embeded ℓ-dimensional manifolds (ℓ≥2).