arXiv research

New method shows nonorientable surfaces in 4D are topologically unknotted.

Golden ratio found on odd genus nonorientable surfaces.

Minimal singular fibers found in nonorientable Lefschetz fibrations.

Study on maximal surfaces with high genus in Lorentz-Minkowski space.

Study compares nonorientable genus values of torus knots.

The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to the orientable four-ball genus, which is bounded below by the Murasugi signature, the Ozsvath-Szabo tau-invariant, the Rasmussen s-invariant, the best lowe…

Two elements generate all mappings of a nonorientable surface.

New bounds on nonorientable four-ball genus for torus knots.

Study on nonorientable 4-genus of double twist knots.

New findings on generating mapping class groups of nonorientable surfaces.

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

Geography problem for nonorientable surfaces bounded by knots.

We prove that the hyperelliptic mapping class group of a nonorientable surface of genus $g\geq 4$ has a faithful linear representation of dimension $g^2-1$ over $\mathbb{R}$.

Maximal dilatation found on nonorientable surfaces.

Let $N$ be a connected nonorientable surface of genus $g$ with $n$ punctures. Suppose that $g$ is odd and $g+n \geqslant 6$. We prove that the automorphism group of the complex of curves of $N$ is isomorphic to the mapping class group $\mathcal M_{N}$ of $N$.

The nonorientable 4-genus $γ_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $γ_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjec…

Automorphism group of nonorientable surface curve graph matches surface homeomorphisms.

Generators found for nonorientable surfaces with many punctures.

We found an infinite family of counterexamples to Batson's conjecture.

For nonorientable surfaces, curve complexes can be exhausted by finite superrigid sets.

The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.

New generators found for twist subgroup of nonorientable surfaces.

The study explores nonorientable 3-manifolds using open books and their monodromies.

This paper extends quasimorphism results to nonorientable surfaces.

Study exhausts curve complex on nonorientable surfaces using finite rigid sets.

We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus $g\ge 3$. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

Klein bottle embeds into specific lens spaces.

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus, orientability, and aggregate slope that a surface spanning that link can have. To thi…

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus $g\ge 7$ is generated by conjugates of one cross…

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.

Proves mapping class group of nonorientable surfaces can be generated by three torsions.

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $λ: \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $λ$ is induced by a homeomorphism …

Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $(g, n) \in \{(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\}$ or $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary…

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a clos…

Nonorientable 4-manifolds can be fibred over 2-disks with nonorientable fibers.

Researchers found that the twist subgroup can be generated by two elements for certain surface genera.

Survey of minimal generating sets for nonorientable mapping class groups.

Let $N_{g,s}$ denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group $M(N_{g,s})$ of the surface $N_{g,s}$, where $s\in{0,1}$ and $g+s>3$. Following this work we obtain a finite presentation for the sub…

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admi…

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $λ:\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightar…

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.

Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class…

We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $g\ge4$.

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal t…

New concordance invariant from spectral sequence on Khovanov homology.