Representativity and waist of cable knots

Waist and Trunk of Knots

We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist of all closed incompressible surfaces in the complement of the knot. On the o...

Cluster Characterization through a Representativity Measure

Clustering is an unsupervised learning task which provides a decomposition of a dataset into subgroups that summarize the initial base and give information about its structure. We propose to enrich this result by a numerical coefficient that describes the cluster representativity and indicates the extent to which they are characteristic of the whole dataset. It is defined for a specific cluster...

A simple construction of high representativity triangulations

Bridge Position and the Representativity of Spatial Graphs

First, we extend the Otal’s result for the trivial knot to the trivial spatial graph, namely, we show that for any bridge tangle decomposing sphere S for a trivial spatial graph Γ, there exists a 2-sphere F such that F contains Γ and F intersects S in a single loop. Next, we introduce two invariants for spatial graphs. As a generalization of the bridge number for knots, we define the bridge str...

Representativity and Complementarity in Tai Chi as Embodied Documentation