Homologically visible closed geodesics on complete surfaces

In this paper, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.

Möbius Band Surprise: A Systematic Illusion in Imagery

When a Möbius loop is cut along the middle of the band, the result is a single connected loop, yet anecdotal evidence from science demonstrations and the use of this effect in magic tricks suggest that most people are thoroughly surprised by this because they strongly believe that the result should be two separate loops. Here, we present results from a behavioral experiment confirming this anecdotal evidence and discuss potential theoretical explanations for why this demonstration evokes strong, but misleading intuitions and a related illusion of impossibility.

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

<p style='text-indent:20px;'>We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.</p><p style='text-indent:20px;'>Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-map (for <inline-formula><tex-math id="M2">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula> large) of a non-transitive Anosov flow <inline-formula><tex-math id="M3">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula> on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to <inline-formula><tex-math id="M4">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula>. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.</p>

Chiral smoothings of knots

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

Explore, on a Mobius strip

“Explore, on a Mobius strip” offers an introduction to the mathematical subfield of topology by way of numerous hand-drawn sketches and an accessible discussion of going for a “walk” on a one-sided, one-edged Mobius strip—also known as a Mobius band. The chapter provides directions for making a Mobius strip out of paper and examining its mathematical properties. Mathematics students and enthusiasts are encouraged to explore more in both mathematics and life in order to expand their worldview. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Homotopy properties of smooth functions on the Möbius band

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on non-orientable compact surfaces $N$.

A note on O. Frolkina’s paper “pairwise disjoint moebius bands in space”

We note a simple algebraic proof of Frolkina’s result that [Formula: see text] does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.