Nonorientable Surface Of Genus Research Articles

Excluded minors for the Klein bottle II. Cascades

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained.

RELAXING MULTICURVES ON THE TWICE PUNCTURED MÖBIUS BAND

Let N1,n (n ≥ 1) be a non-orientable surface of genus 1 with n punctures and one boundary component. Generalized Dynnikov coordinates provide a bijection between the set of multicurves in N1,n and Z2n−1 \ {0}. In this paper we restrict to the case where n = 2 and describe an algorithm to relax a multicurve in N1,2 making use of its generalized Dynnikov coordinates

Curves on Non-Orientable Surfaces and Crosscap Transpositions

Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.

Connected components of Isom(varvec{pmb {mathbb {H}}^3})-representations of non-orientable surfaces

Let N_k denote the closed non-orientable surface of genus k. In this paper we study the behaviour of the ‘square map’ from the group of isometries of hyperbolic 3-space to the subgroup of orientation preserving isometries. The properties of the ‘square map’ and other related maps serve as a technical step towards the counting of the connected components of the variety of representations of pi _1(N_k) in Isom(mathbb {H}^3). We show that the variety of representations hom(pi _1(N_k),mathrm {Isom}(mathbb {H}^3)) has 2^{k+1} connected components, which are distinguished by the Stiefel-Whitney classes of the associated flat bundle.

CONNECTEDNESS OF THE CUT SYSTEM COMPLEX ON NONORIENTABLE SURFACES

Let N be a compact, connected, nonorientable surface of genus g with n boundary components. In this note, we show that the cut system complex of N is connected for g < 4 and disconnected for g ≥ 4. We then define a related complex and show that it is connected for g ≥ 4.

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.

Thin position through the lens of trisections of 4-manifolds

Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all manifolds having width equal to $\{1,\dots, 1\}$, and give a relation between the width of $M$ and its double $M\cup_{id_\partial} \overline M$. In particular, we describe how to obtain genus $2g+2$ and $g+2$ trisection diagrams for sphere bundles over orientable and non-orientable surfaces of genus $g$, respectively. By last, we study the problem of describing relative handlebodies as cyclic covers of 4-space branched along knotted surfaces from the width perspective.

Surface codes and color codes associated with non-orientable surfaces

Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

Exhausting curve complexes by finite superrigid sets on nonorientable surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal {C}(N)$ be the curve complex of $N$. We prove that if $(g, n) \neq (1,2)$ and $g + n \neq 4$, then there is an exhaustion of $\mathcal {C}(N)$

A NOTE ON CHAINS AND BOUNDING PAIRS OF DEHN TWISTS

AbstarctLet $N_g^k$ be a nonorientable surface of genus g with k punctures. In the first part of this note, after introducing preliminary materials, we will give criteria for a chain of Dehn twists to bound a disc. Then, we will show that automorphisms of the mapping class groups map disc bounding chains of Dehn twists to such chains. In the second part of the note, we will introduce bounding pairs of Dehn twists and give an algebraic characterization for such pairs.

Superinjective simplicial maps of the two-sided curve complexes on nonorientable surfaces

Let N be a compact, connected, nonorientable surface of genus g >= 5 with n >= 0 boundary components. Let T(N) be the two-sided curve complex of N. If lambda: T(N) -> T(N) is a superinjective simplicial map, then there exists a homeomorphism h: N -> N unique up to isotopy such that H(alpha) = lambda(alpha) for every vertex alpha in T(N) where H = [h].

Trajectory equivalence of optimal Morse flows on closed surfaces

We consider optimal Morse flows on closed surfaces. Up to topological trajectory equivalence such flows are determined by marked chord diagrams. We present list all such diagrams for flows on nonorientable surfaces of genus at most 4 and indicate pairs of diagrams corresponding to the flows and their inverses.

Injective homomorphisms of mapping class groups of non-orientable surfaces

Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0, and let M(N) be the mapping class group of N. We show that, if G is a finite index subgroup of M(N) and ϕ : G → M(N) is an injective homomorphism, then there exists f ∈ M(N) such that ϕ(g) = f gf^( -1) for all g ∈ G. We deduce that the abstract commensurator of M(N) coincides with M(N).

Minimum cycle and homology bases of surface-embedded graphs

We study the problems of finding a minimum cycle basis (a minimum-weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum-weight set of cycles that generates the 1-dimensional $(\mathbb{ Z}_ 2 )$-homology classes) of an undirected graph cellularly embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $ O ( n^ ω + 2^{ 2 g} n^ 2 + m )$-time algorithm for graphs cellularly embedded on an orientable surface of genus $ g$ . Prior to this work, the best known algorithms for surface-embedded graphs were those for general graphs: an $ O ( m^ ω )$-time Monte Carlo algorithm and a deterministic $ O ( nm^ 2 / log + n^ 2 m )$-time algorithm . For the minimum homology basis problem, we give a deterministic $ O (( g + b )^ 3 n log + m )$-time algorithm for graphs cellularly embedded on an orientable or non-orientable surface of genus $ g$ with $ b$ boundary components, improving on existing algorithms for many values of $ g$ and $ n$ . The algorithm assumes that shortest paths are unique; this assumption can be avoided by either using random perturbations of the edge weights guaranteeing a high probability of success or by deterministic means at a cost of an $ O (\log )$ factor increase in running time.

Non-existence of polyhedral immersions of triangulated surfaces in $$\mathbb R^3$$ R 3

We present and apply a method for disproving the existence of polyhedral immersions in \(\mathbb R^3\) of certain triangulations on non-orientable surfaces. In particular, it is proved that neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as immersed polyhedral surfaces in \(\mathbb R^3\).

A presentation for the mapping class group of a nonorientable surface

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 $g+n>3$.

On finite index subgroups of the mapping class group of a nonorientable surface

Let $M(N_{h,n})$ denote the mapping class group of a compact nonorientable surface of genus $h\ge 7$ and $n\le 1$ boundary components, and let $T(N_{h,n})$ be the subgroup of $M(N_{h,n})$ generated by all Dehn twists. It is known that $T(N_{h,n})$ is the unique subgroup of $M(N_{h,n})$ of index $2$. We prove that $T(N_{h,n})$ (and also $M(N_{h,n})$) contains a unique subgroup of index $2^{g-1}(2^g-1)$ up to conjugation, and a unique subgroup of index $2^{g-1}(2^g+1)$ up to conjugation, where $g=\lfloor(h-1)/2\rfloor$. The other proper subgroups of $T(N_{h,n})$ and $M(N_{h,n})$ have index greater than $2^{g-1}(2^g+1)$. In particular, the minimum index of a proper subgroup of $T(N_{h,n})$ is $2^{g-1}(2^g-1)$.

An improved upper bound for the bondage number of graphs on surfaces

The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph G with maximum degree Δ(G) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, b(G)≤min{Δ(G)+h+2,Δ+k+1}. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of h and k. In this paper we establish an improved explicit upper bound for b(G), using the Euler characteristic χ instead of the genera h and k, with the relations χ=2−2h and χ=2−k. We show that b(G)≤Δ(G)+⌊r⌋ for the case χ≤0 (i.e. h≥1 or k≥2), where r is the largest real root of the cubic equation z3+2z2+(6χ−7)z+18χ−24=0. Our proof is based on the technique developed by Carlson–Develin and Gagarin–Zverovich, and includes some elementary calculus as a new ingredient. We also find an asymptotically equivalent result b(G)≤Δ(G)+⌈12−6χ−1/2⌉ for χ≤0, and a further improvement for graphs with large girth.

DOUBLES OF KLEIN SURFACES

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

Upper bounds for the bondage number of graphs on topological surfaces

The bondage number b(G) of a graph G is the smallest number of edges of G whose removal results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree Δ(G) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, b(G)≤min{Δ(G)+h+2,Δ(G)+k+1}. This generalizes known upper bounds for planar and toroidal graphs, and can be improved for bigger values of the genera h and k by adjusting the proofs.