On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

Exhausting curve complexes by finite rigid sets on nonorientable surfaces

Let [Formula: see text] be a compact, connected, nonorientable surface of genus [Formula: see text] with [Formula: see text] boundary components. Let [Formula: see text] be the curve complex of [Formula: see text]. We prove that if [Formula: see text] or [Formula: see text], then there is an exhaustion of [Formula: see text] by a sequence of finite rigid sets. This improves the author’s result on exhaustion of [Formula: see text] by a sequence of finite superrigid sets.

A Symplectic Dynamics Proof of the Degree–Genus Formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

Determining the doubly slice genera of prime knots with up to 12 crossings

For a knot [Formula: see text], the doubly slice genus [Formula: see text] is the minimal [Formula: see text] such that [Formula: see text] divides a closed, orientable, and unknotted surface of genus [Formula: see text] embedded in [Formula: see text]. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.

Stability of symmetric powers of vector bundles of rank two with even degree on a curve

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

Tied links in various topological settings

Tied links in [Formula: see text] were introduced by Aicardi and Juyumaya as standard links in [Formula: see text] equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces [Formula: see text], in handlebodies of genus [Formula: see text], and in the complement of the [Formula: see text]-component unlink. We introduce the tied braid monoids [Formula: see text] by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an [Formula: see text]-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

Numerical semigroups of small and large type

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

Kauffman bracket skein module of the connected sum of handlebodies and non-injectivity

For the handlebody [Formula: see text] of genus [Formula: see text], Przytycki studied the (Kauffman bracket) skein module [Formula: see text] of the connected sum [Formula: see text] at [Formula: see text]. One of his results is that, in the case when [Formula: see text] is invertible for any [Formula: see text], a homomorphism [Formula: see text] is an isomorphism, which is induced by a natural way. In this paper, in the case when [Formula: see text], the ground ring is [Formula: see text], and [Formula: see text] is a [Formula: see text]-th root of unity ([Formula: see text]), we show that [Formula: see text] is not injective.