Narrow Results

Given a tangle diagram of a classical knot K, we construct a surface diagram of any twist-spun 2-knot of K. From the crossings of the tangle diagram, we can get information of the corresponding triple points of the surface diagram, which are used to compute cocycle invariants of twist-spun 2-knots.

We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can take.

In this paper, we prove that if a surface diagram of a surface-knot has at most two triple points and the lower decker set is connected, then the surface-knot group is isomorphic to the infinite cyclic group.

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exist a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask that whether any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained.

We use a triple-point version of the Whitney trick to show that ornaments of three orientable $(2k-1)$-manifolds in ${ℝ}^{3k-1}$, $k>2$, are classified by the $\mu$-invariant.

A very similar (but not identical) construction was found independently by I. Mabillard and U. Wagner, who also made it work in a much more general situation and obtained impressive applications. The present note is, by contrast, focused on a minimal working case of the construction.