Paper classifies link diagrams on nonorientable surfaces using region crossing changes.
problem Classifying link diagrams on nonorientable surfaces.
method Classification through region crossing changes.
result Classification of link diagrams on nonorientable surfaces.
New method shows nonorientable surfaces in 4D are topologically unknotted.
problem Tackles isotopy classes of nonorientable surfaces in D4. method Calculations implemented in Sage to show ambient isotopy.
result Closed, nonorientable surfaces in S4 are topologically unknotted. New stability theorem for nonorientable surfaces mapping class groups.
problem Stability of homology groups of mapping class groups of nonorientable surfaces.
method Galatius--Kupers--Randal-Williams framework of cellular E2-algebras. result New best known stability range for homology of nonorientable surfaces.
Study on nonorientable 4-manifolds using simplified fibrations and trisections.
problem Classify and understand nonorientable 4-manifolds.
method Use simplified broken Lefschetz fibrations and trisections, topological modifications of singularities, handlebody decompositions, and mapping classes of surfaces.
result Classify low genus simplified broken Lefschetz fibrations on nonorientable 4-manifolds.
Study shows pants graph automorphisms match mapping class groups of nonorientable surfaces.
problem Understanding automorphisms of pants graphs on nonorientable surfaces.
method Analyzing mapping class groups and proving isomorphism.
result Automorphism group of pants graphs isomorphic to mapping class groups.
Minimal singular fibers found in nonorientable Lefschetz fibrations.
problem Finding the minimal number of singular fibers in nonorientable Lefschetz fibrations.
method Analyzing admissible nonorientable genus g Lefschetz fibrations over orientable surfaces.
result Existence of one singular fiber if and only if g ≥ 4 and h ≥ 1.
Maximal dilatation found on nonorientable surfaces.
problem Finding maximal dilatation on nonorientable surfaces.
method Proving irreducibility of a polynomial to show maximal dilatation.
result Maximal dilatation is achieved by the Liechti-Strenner polynomial.
We construct complete nonorientable minimal surfaces whose Gauss map omits two points of the projective plane. This result proves that Fujimoto's theorem is sharp in nonorientable case.
Generators found for nonorientable surfaces with many punctures.
problem Identifying minimal generating sets for mapping class groups of nonorientable surfaces.
method Analyzing extrmMod(Ng,p) for g≥14 to find generator counts. result Generators of extrmMod(Ng,p) can be as few as 5 or 6. Two elements generate all mappings of a nonorientable surface.
problem Generating the mapping class group of a nonorientable surface.
method Proving two elements generate the mapping class group for g≥13. result The mapping class group of a nonorientable surface of genus g≥13 can be generated by exactly two elements. Minimal involutions generate a subgroup of nonorientable surfaces.
problem Generating a minimal set of involutions for a specific subgroup.
method Obtained a minimal generating set of involutions.
result Minimal involutions for the level 2 subgroup of a nonorientable surface.
This paper extends quasimorphism results to nonorientable surfaces.
problem Understanding quasimorphisms on nonorientable surface diffeomorphism groups.
method Constructing infinitely many quasimorphisms on the identity component of nonorientable surface diffeomorphism groups.
result The space of nontrivial quasimorphisms on the identity component of the diffeomorphism group of a closed nonorientable surface is infinite-dimensional.
Nonorientable 4-manifolds can be fibred over 2-disks with nonorientable fibers.
problem Fibering nonorientable 4-manifolds over 2-disks.
method Constructing Lefschetz fibrations over 2-disks with nonorientable fibers.
result Every nonorientable closed 3-manifold admits an open book decomposition.
Study of alternating links on nonorientable surfaces, extending results to nonorientable projections.
problem Generalizing hyperbolic geometry results to nonorientable surfaces.
method Extending results from orientable to nonorientable surfaces.
result Klein-bottly alternating links in prism manifolds have hyperbolic geometry.
Zeta functions extended to nonorientable surfaces, order of vanishing computed.
problem Computing dynamical zeta functions for nonorientable surfaces.
method Simple argument extending microlocal proofs to nonorientable case.
result Order of vanishing of zeta function is the first Betti number.
Geography problem for nonorientable surfaces bounded by knots.
problem Bounding and computing the nonorientable 4-genus of knots.
method Analysis of existing methods, relationships between Betti number and normal Euler class, exploration of families of torus knots, use of Ozsváth-Szabó d-invariant.
result Improvement on the bound for some knots using the Upsilon invariant.
New findings on generating mapping class groups of nonorientable surfaces.
problem Understanding the minimum number of elements needed to generate the mapping class group of nonorientable surfaces.
method Proving the minimum number of generators for extrmMod(Ng) for g≥19 and g≥26. result For g≥19, extrmMod(Ng) can be generated by two elements, one of order g. For g≥26, extrmMod(Ng) can be generated by three involutions. New bounds on nonorientable four-ball genus for torus knots.
problem Finding bounds on the nonorientable four-ball genus of torus knots.
method Combining knot Floer homology techniques to derive lower bounds.
result Sharp bounds for several families of torus knots, including T4n,(2n±1)2. 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…
Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.
problem Periodic knots do not always have nonorientable spanning surfaces of high genus.
method Examples and calculations of nonorientable spanning surfaces of periodic knots.
result The first Betti number of nonorientable spanning surfaces can be arbitrarily large.
Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.
problem Tackles the unknotting of nonorientable surfaces in 4-spheres and 4-balls.
method Uses topological isotopy and properties of knot groups and normal Euler numbers.
result Proves that certain nonorientable surfaces are topologically unknotted under specific conditions.
In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over …
New classification of nonorientable 4-manifolds with specific fundamental groups.
problem Classifying nonorientable 4-manifolds with cyclic fundamental groups.
method Simple cut-and-paste construction, using results from Hambleton-Kreck-Teichner and Khan.
result Plausible classification of a large set of nonorientable 4-manifolds with cyclic fundamental groups of order 2p.
Study on nonorientable 4-genus of double twist knots.
problem Determining the nonorientable 4-genus of double twist knots.
method Explicit constructions and obstructions from Donaldson's diagonalization theorem.
result Proved bounds on nonorientable 4-genus for infinite subfamilies of double twist knots.
Uniform hyperbolicity proved for nonorientable surface curve graphs.
problem Proving uniform hyperbolicity for nonorientable surface curve graphs.
method Using bicorn curves and arguments from orientable surfaces.
result Graph of nonseparating curves is uniformly hyperbolic.
On each nonorientable surface of odd genus g≥5, we give a mapping class whose dilatation on an invariant subsurface is the golden ratio.
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…
Nonorientable surface mapping class group embeds quasi-isometrically in its orientable cover.
problem Embedding nonorientable surface mapping class group in orientable surface mapping class group.
method Utilized semihyperbolicity of orientable surface mapping class group and orientation double covering properties.
result Injective homomorphism is a quasi-isometric embedding.
The nonorientable 4-genus γ4(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 γ4 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.
problem Identifying automorphisms of nonorientable surface curve graphs.
method Using Bowden, Hensel, and Webb's fine curve graph and Long, Margalit, Pham, Verberne, and Yao's proof as a foundation.
result Automorphism group of nonorientable surface curve graph is isomorphic to the surface's homeomorphism group.
The study bounds the excess of disjoint nonorientable surfaces in a 4-manifold.
problem Bounding the excess of disjoint nonorientable surfaces in a 4-manifold.
method Combining tubing construction with signature and Euler-characteristic formulas for 2-fold branched covers.
result The normal-Euler excess is bounded by a constant depending only on the ambient 4-manifold.
We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.
Proves mapping class group of nonorientable surfaces can be generated by three torsions.
problem Generates mapping class group of nonorientable surfaces by three torsions.
method Proves using algebraic methods and properties of nonorientable surfaces.
result Proves mapping class group of nonorientable surfaces can be generated by three torsions.
Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties o…
We determine the first homology group of the mapping class group M(N) of a nonorientable surface N with coefficients in H_1(N;Z).
Survey of minimal generating sets for nonorientable mapping class groups.
problem Challenges in generating minimal sets for nonorientable surfaces.
method Detailed analysis of various generating sets, including torsions, involutions, and commutators.
result For large genus, both Mod(Ng) and Tg are generated by two elements. We show that Norbury's McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of pi_1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.
Klein bottle embeds into specific lens spaces.
problem Embeddability of Klein bottle in lens spaces.
method Direct proof and explicit realizations.
result Klein bottle embeds into L(4n,2n±1) only. 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≥7 is generated by conjugates of one cross…
Embedding right-angled Artin groups in mapping class groups of nonorientable surfaces.
problem Embedding right-angled Artin groups in mapping class groups for nonorientable surfaces.
method Proving embedding for finite full subgraphs and some non-full subgraphs of curve graphs.
result Existence of non-full subgraphs that can be embedded in mapping class groups.
We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let T(p,q) be any torus knot with p even and q odd. The difference between these two invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and k≥0. Hence, the…
We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.
We obtain a simple presentation of the hyperelliptic mapping class group Mh(N) of a nonorientable surface N. As an application we compute the first homology group of Mh(N) with coefficients in H1(N;Z).
Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
problem Classifying homomorphisms from braid groups to mapping class groups of nonorientable surfaces.
method Classifies homomorphisms based on the properties of Dehn twists and crosscap transpositions.
result Every homomorphism is either cyclic or maps generators to distinct Dehn twists or crosscap transpositions.
We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus g≥3. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.
We prove that the hyperelliptic mapping class group of a nonorientable surface of genus g≥4 has a faithful linear representation of dimension g2−1 over R.
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…