Narrow Results

In the present paper, we will study the creation of Klein bottles by Dehn surgery on knots in the 3-sphere, and we will give an upper bound for slopes creating Klein bottles for non-cabled knots by using the genera of knots. In particular, it is shown that if a Klein bottle is created by Dehn surgery on a genus one knot then the knot is a doubled knot. As a corollary, we obtain that genus one, cross-cap number two knots are doubled knot.

This paper is a survey paper that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their 3-dimensional projections, and indeed the illustrations here depend upon a further projection into the plane of the page. The concepts used to develop the illustrations will be developed herein.

Let $K$ be a hyperbolic knot in the $3$-sphere. If a $p/q$-Dehn surgery on $K$ produces manifold with an embedded Klein bottle or essential $2$-torus, then we prove that $\left|p\right|\le 8g_K-3$, where $g_K$ is the genus of $K$. We obtain different upper bounds according to the production of a Klein bottle, a non-separating $2$-torus, or an essential and separating $2$-torus. The well known examples which are the figure eight knot and the pretzel knot $K(-2,3,7)$ reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed $3$-manifolds.

We will prove that if some Dehn surgery on a composite knot in the 3-sphere creates a 3-manifold containing a Klein bottle then the knot is a connected sum of two 2-cabled knots.

Originally developed as an exhibition for the Providence Art Clubs' Dodge Gallery in March of 1996, "Surface Beyond the Third Dimension" continues its existence today as a virtual experience on the internet. Here, the authors take you on a tour of the exhibit and describe both the artwork and the mathematics underlying the images, whose subjects range from projections of spheres and tori in four-space, to images of complex functions, to views of the Klein bottle. Along the way, the authors introduce some of the issues that arise from this dual presentation approach, and point out some of the enhancements available in the virtual gallery that are not possible in the physical one.

My purpose here is to give some background to a series of sculptures I have been creating in which nonorientable surfaces, especially crosscaps, play a mathematical content role. The fundamental circle of theorems involved is on the classification of surfaces in terms of handles and crosscaps. The most basic examples are spheres, tori, möbius bands, projective planes, and klein bottles, cf., [CF1, HF1] for discussion of my earlier sculpture involving these concepts. For an exposition and further references to the mathematical background for the circle of theorems I refer to, see [FG1].

In this paper, we study an infinite-dimensional Lie algebra ℬ_q, called the q-analog Klein bottle Lie algebra. We show that ℬ_q is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬ_q are also determined.

The product S³ × S⁵ of 3-dimensional and 5-dimensional spheres is a complex manifold, which is defined by Calabi-Eckmann ([2]). This complex structure comes from the fact that S³ ×S⁵ is a total space of a T²-bundle over ℂℙ¹ ×ℂℙ², with a natural projection map π₁ × π₂ : S³ × S⁵→ ℂℙ¹ × ℂℙ². Let $W_1=ℂ{\mathbb{P}}^2\text{\#}\overline{ℂ{\mathbb{P}}^2}$ be one of Hirzebruch surfaces which is obtained as a blowing up of ℂℙ² at one point. This surface realizes as a complex submanifold in ℂℙ¹ ×ℂℙ². Then the inverse image ${\tilde{W}}_1={\left({\pi }_1\times {\pi }_2\right)}^{-1}\left(W_1\right)$ of W₁ is a complex submanifold in S³ × S⁵. First we shall show that ${\tilde{W}}_1$ is diffeomorphic to S³ × S³ concretely. By using this diffeomorphism, we give a representation of the induced complex structure J of ${\tilde{W}}_1$ from S³ × S⁵, as a (1, 1) tensor field on ${\tilde{W}}_1$ with respect to the left invariant vector fields on S³ × S³. We note that the induced complex structure J is different from the (Calabi-Eckmann type) complex structure of a direct Riemannian product S³ × S³, which is obtained as a total space of a T² bundle over ℂℙ¹ × ℂℙ¹. Also we describe a blowing up of ℝℙ² at one point, as a Lagrangian surface of W₁. Then this Lagrangian surface is a Klein bottle, which can be considered as a connected sum ℝℙ²#ℝℙ² in $ℂ{\mathbb{P}}^2\text{\#}\overline{ℂ{\mathbb{P}}^2}$.

Let F be an embedded Klein bottle in S⁴\{∞}. If the singular set Γ(F*) of the projection F* of F into R³ consists of at most three disjoint simple dosed curves, then F bounds a said Klein bottle in the 4-sphere S⁴, i.e., F can be moved to the standard Klein bottle.