Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, $$\mathfrak{m}$$ m (2-Knaster), $$\mathfrak{p}$$ p , $$\mathfrak{h}$$ h , the splitting number $$\mathfrak{s}$$ s and the reaping number $$\mathfrak{r}$$ r .

Links with splitting number one

We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can therefore also determine whether the unlinking number of L is 1.

Slice-torus Concordance Invariants and Whitehead Doubles of Links

In this paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent of the corresponding slice-torus link invariant.

Connected Numbers and the Embedded Topology of Plane Curves

The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

Splitting number of augmented links

We completely determine the splitting number of augmented links arising from knot and link diagrams in which each twist region has an even number of crossings. In the case of augmented links obtained from knot diagrams, we show that the splitting number is given by the size of a maximal collection of Boromean sublinks, any two of which have one component in common. The general case is stablished by considering the linking numbers between components of the augmented links. We also discuss the case when the augmented link arises from a link diagram in which twist regions may have an odd number of crossings.

Splitting Numbers of Links

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.