Journal of Knot Theory and Its Ramifications

We completely classify the topological and geometric structures of some series of closed connected orientable 3-manifolds introduced by Kim and Kostrikin in [20,21] as quotient spaces of certain polyhedral 3-cells by pairwise identifications of their boundary faces. Then we study further classes of closed orientable 3-manifolds arising from similar polyhedral schemata, and describe their topological properties.

The inverse braid monoid consists of n-string braids from which some subset of the strings has been deleted. Such partial braids may be composed by concatenation followed by the deletion of any incomplete strings, and this operation gives the structure of an inverse monoid. We derive presentations of by first using a groupoid presentation, and then deriving equivalent inverse monoid presentations. In particular, we offer a new derivation of the presentation for that has been given by Easdown and Lavers.

The nth hull of a union of curves in ℝ³ is the set of points with the property: Any plane passing through the point intersects the curves at least 2n times. The hull number of a link L is defined as the minimum number of non-empty hulls a representative of L can have. We show that the hull numbers of torus links is smaller than expected, but still large. In particular, for a torus link of type (p, p) hull number equals , and in general for the type (p, q) with q > p it is greater than or equal to .

We explain an algorithm for finding a boundary link Seifert matrix for a given multivariable Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.

An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete knot is the digitalized representation of a knot, which is composed of the minimum number of constant orthogonal straight-line segments and is represented by the knot-number notation. The proposed algorithm for obtaining minimal discrete knots tries to reduce the number of straight-line segments of a given discrete knot preserving its shape. This algorithm is based on two fundamental transformations: U and L.

In order to prove the efficiency and rapidity of the algorithm, a great variety of examples of discrete knots are presented: complete families of discrete knots at different orders; random discrete knots; examples of non-trivial and unsolved discrete knots.

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

A two-component link with nonzero even linking number cannot be equivalent to its mirror image by an orientation preserving map that sends each component to its own mirror image.