Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

A characterization of alternating links in thickened surfaces

We use an extension of Gordon–Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\Sigma \times I$ , then the Gordon–Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$ . A compact surface in $\Sigma \times I$ is called definite if its Gordon–Litherland pairing is a definite form. We prove that a link $L$ in a thickened surface is non-split, alternating, and of minimal genus if and only if it bounds two definite surfaces of opposite sign.

On s-extremal Riemann surfaces of even genus

We consider Riemann surfaces of even genus g with the action of the group $$\mathcal {D}_n\times \mathbb {Z}_2$$ D n × Z 2 , with n even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called s-extremal. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an s-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called extremal. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.

Rational curves on genus-one fibrations

In this paper we look for necessary and sufficient conditions for a genus-one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus-one fibration $$X\rightarrow B$$ X → B does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product $$\tilde{B}\times E$$ B ~ × E over B. Many sufficient conditions for the existence of rational curves in a variety that admits a genus-one fibration are proved in this paper.

Rational curves on fibered varieties

Let X be a complex projective variety with log terminal singularities and vanishing augmented irregularity. In this paper, we prove that if X admits a relatively minimal genus-one fibration, then it contains a subvariety of codimension one covered by rational curves contracted by the fibration. We then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result.

Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms

We study the symmetric Riemann surfaces for which the group of orientation preserving automorphisms acts without fixed points. We show that any finite group can give rise to such an action, determine the maximal number of non-conjugate symmetries for such surfaces and find a sharp upper bound on maximal total number of ovals for a set of $k$ symmetries with ovals. We also solve the minimal genus problem for dihedral groups acting on the surfaces described above, for odd genera.

Minimal partitions for critical Heegaard splittings

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of [Formula: see text], which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition.

A distance for circular Heegaard splittings

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.