Journal of Knot Theory and Its Ramifications

This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely when there is no filamentation of the knot, as described by Hrencecin and Kauffman. Schellhorn has provided a version of filamentations for flat virtual links. Our polynomial has the property that if there is a filamentation, then the polynomial is 0. The converse fails, although if the polynomial is 0, then it turns out to be easy to determine if there is a filamentation.

A p-periodic 3-manifold is a 3-manifold that admits a ℤ_p-action whose fixed point set is a circle. We give a congruence that relates the quantum invariant of a p-periodic 3-manifold associated to any modular category over an integrally closed ground ring and the corresponding quantum invariant of its orbit space.

We show that a knot in S³ with an infinite number of incompressible Seifert surfaces contains a closed non-peripheral incompressible surface in its complement.

Let K be a knot or link and let p = det(K). Using integral colorings of rational tangles, Kauffman and Lambropoulou showed that every rational K has a mod p coloring with distinct colors. If p is prime this holds for all mod p colorings. Harary and Kauffman conjectured that this should hold for prime, alternating knot diagrams without nugatory crossings for p prime. Asaeda, Przyticki and Sikora proved the conjecture for Montesinos knots. In this paper, we use an elementary combinatorial argument to prove the conjecture for prime alternating algebraic knots with prime determinant. We also give a procedure for coloring any prime alternating knot or link diagram and demonstrate the conjecture for non-algebraic examples.

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

We introduce a method to estimate the virtual crossing number of a 2-component flat virtual link. In particular, we give an estimation for the case that a surface realization of the link is expressed by two non-homologous simple loops on the torus (whose supporting genus is one automatically), and determine for some special cases. To know estimations, it is important to consider "realizing problem".