Journal of Knot Theory and Its Ramifications

A generic map from a finite graph to the 2-space is called identifiable if any two embeddings of the graph into the 3-space obtained by lifting the map with respect to the natural projection from the 3-space to the 2-space are ambient isotopic in the 3-space. We show that only planar graphs have identifiable maps. We characterize the identifiable maps for some planar graphs.

Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle).

A self-dual definition of a Frobenius object without the use of a prefered dual is considered.

The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors.

Hermitian structures are modeled as a conjugation compatible with duality. It is the structure cobordism categories posses. A definition of generalized cobordism categories is proposed.

We introduce topological helicity, an invariant for oriented framed links. Topological helicity provides an elementary means of computing helicity for a magnetic flux rope by measuring its knotting, linking, and twisting. We present an equivalence relation, reconnection-equivalence, for framed links and prove that topological helicity is a complete invariant for the resulting equivalence classes. We conclude by showing that one can use magnetic reconnection to transform one collection of linked flux ropes into another collection if and only if they have the same helicity.

In [2], it was shown that every spatial embedding of K₁₀, the complete graph on ten vertices, contains a non-split 3-component link (K₁₀ is intrinsically 3-linked). We improve this result by exhibiting two different subgraphs of K₁₀ that also have this property. In addition, we also exhibit several families of graphs that are intrinsically 3-linked.

There are some phenomena arising in the virtual knot theory which are not the case for classical knots. One of them deals with the "breaking" procedure of knots and obtaining long knots. Unlike the classical case, they might not be the same. The present work is devoted to construction of some invariants of long virtual links. Several explicit examples are given. For instance, we show how to prove the non-triviality of some knots obtained by breaking virtual unknot diagrams by very simple means.

In this paper, we study lens knots and periodic knots by using integral Vassiliev invariants. Knot polynomials such as the Jones, HOMFLY, Kauffman polynomials give infinitely many integral Vassiliev invariants and we get some necessary conditions for a link to be a lens knot or a periodic link by using these polynomial invariants.

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal ℙ²-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal ℙ²-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.

We consider a family of words in a free group of rank n which determine 3-manifolds ℳ_n(p,q). We prove that the fundamental groups of ℳ_n(p,q) are cyclically presented, and that ℳ_n(p,q) is the n-fold cyclic covering of the 3-sphere branched over the torus knots T(p,q) if p is odd and q≡±2(mod p). We also obtain an explicit Dunwoody parameters for the torus knots T(p,q) for odd p and q≡±2(mod p).