No abstract

A weave is the lift to the Euclidean thickened plane of a set of infinitely many planar crossed geodesics, that can be characterized by a number of sets of threads describing the organization of the non-intersecting curves, together with a set of crossing sequences representing the entanglements. In this paper, the classification of a specific class of doubly periodic weaves, called untwisted (p, q)-weaves, is done by their crossing number, which is the minimum number of crossings that can possibly be found in a unit cell of its infinite weaving diagrams. Such a diagram can be considered as a particular type of quadrivalent periodic planar graph with an over or under information at each vertex, whose unit cell corresponds to a link diagram in a thickened torus. Moreover, considering that a weave is not uniquely defined by its sets of threads and its crossing sequences, we also specify the notion of equivalence classes by introducing a new parameter, called crossing matrix.

A weave is the lift to the thickened Euclidean plane of a particular type of quadrivalent planar connected graph with an over or under crossing information to each vertex and such that the lifted components are non-intersecting simple open curves. In this paper, we introduce a formal topological definition of weaves as three-dimensional entangled structures and characterize the equivalence classes of doubly periodic untwisted [Formula: see text]-weaves by introducing a new invariant, called crossing matrix. Finally, we suggest a combinatorial approach to classify this specific class of weaves by their crossing number.

An [Formula: see text]-branched twist spin is a fibered [Formula: see text]-knot in [Formula: see text] which is determined by a [Formula: see text]-knot [Formula: see text] and coprime integers [Formula: see text] and [Formula: see text]. For a [Formula: see text]-knot, Nagasato proved that the number of conjugacy classes of irreducible [Formula: see text]-metabelian representations of the knot group of a [Formula: see text]-knot is determined by the knot determinant of the [Formula: see text]-knot. In this paper, we prove that the number of irreducible [Formula: see text]-metabelian representations of the knot group of an [Formula: see text]-branched twist spin is determined up to conjugation by the determinant of the associated [Formula: see text]-knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin’s presentation of the knot group of the [Formula: see text]-knot.