This month’s topics:
Geometry endangers received wisdom
That is part of what Jordan Ellenberg has to say about geometry in an interview with Anne Strainchamps of Wisconsin Public Radio, broadcast on May 28, 2022. More completely, “Geometry’s power lies in how it endangers received wisdom.” At the center of the interview Ellenberg tells a “parable” (you can hear the excerpt on YouTube) that illustrates his point. It’s the story, from Edwin A. Abbott’s 1884 novella Flatland, about the Square’s encounter with the Sphere. (Flatland is a 2-dimensional universe inhabited by intelligent polygons; Abbott spends several chapters explaining in detail how their society works). Our narrator is the Square, a kind of flat Everyman, who has his life drastically changed by an encounter with a Sphere.
The Sphere explains that his world has another dimension. Initially the Square cannot conceive of an “up” that does not mean “North.” But the Sphere argues by analogy: a point (1 terminal point) moving North produces a segment (2 terminal points) and a segment displaced sideways produces a square (4 terminal points). He remarks that 1, 2, 4 are in geometric progression and asks the Square for the next number. “Eight.” “Exactly.” A square, when displaced “up,” will sweep out what “we call” a Cube, with 8 terminal points. The Square gets involved in the calculation and figures out that since a point has no sides, a segment has 2 “sides” (both points), and a square has 4 sides (all segments), a cube will have to have 6 sides, all squares. Through all of this exposition the Sphere speaks condescendingly to the Square. E.g. “Distress not yourself if you cannot at first comprehend the deeper mysteries of Spaceland.”
Later in the story, after the Sphere has dragged the Square out into Spaceland and shown him how he can see inside all of his fellow polygons, the Square understands, and asks to be taken into into the Fourth Dimension to get the same perspective on the Sphere and the other solids. Now it’s the Sphere’s turn to be obtuse: “There is no such land. The very idea of it is utterly inconceivable.” And here is Ellenberg’s point. As he puts it, “The Square is figuring something out. And the Sphere, who was moments ago in this position of authority is suddenly like, ‘Wait, wait, I don’t like this!'” Geometry is endangering received wisdom.
Perplexing surfaces in Quanta
“Seifert surfaces in the 4-ball” by Kyle Hayden, Seungwon Kim, Maggie Miller, JungHwan Park and Isaac Sundberg was posted on ArXiV on May 30, 2022 and picked up by Josh Harnett for Quanta Magazine on June 16, with the title “Surfaces So Different Even a Fourth Dimension Can’t Make Them the Same.” The team had solved a problem open since 1982 regarding what happens to two topologically equivalent Seifert surfaces of the same knot when their interiors are allowed to move in four dimensions.
A Seifert Surface for a knot is a surface with that knot as boundary.
A knot can have more than one Seifert surface. This happens, for example, when the knot can be drawn without self-intersections on a closed surface so as to cut the surface into two parts. Each of these is a Seifert surface.
The surfaces $\Sigma_0$ and $\Sigma_1$ look quite different, but the difference comes from the way they are embedded in 3-space. Their intrinsic topologies are actually identical. We can verify this by adding vertices and segments so as to cut them up into simple pieces (topologically, polygons) and showing that the resulting cell complexes have the same Euler characteristic.
the two surfaces into topological polygons: $\Sigma_0$ into an octagon and two rectangles, and $\Sigma_1$ into two hexagons and one rectangle. The numbers of vertices, edges and faces can then be used to calculate their Euler characteristics. Each of them has 8 vertices, 12 edges and 3 faces, so they have the same Euler characteristic $V-E+F=-1$.
As the Sphere explained to the Square in our first item, some problems of a topological nature can be resolved by moving to a higher dimension. For example, a point can escape from a circular prison in Flatland by slipping into and out of the third dimension.
Similarly, any knot in 3-space is isotopic to a circle if we are allowed to move some of its points into and out of the fourth dimension. For the problem at hand, many cases were known of a knot with two topologically equivalent Seifert surfaces that are not isotopic in 3-space, but which become isotopic when their interiors are allowed to move in the fourth dimension. The open question was whether or not is this the general situation.
In Harnett’s words, “The new work identifies the first pair of Seifert surfaces that are as provably distinct from each other in four dimensions as they are in three.” $\Sigma_0$ and $\Sigma_1$ are topologically identical Seifert surfaces for the same knot, but even when their interiors are pushed into 4-space, there is no isotopy taking one to the other. So the answer to the question is “no.” How do they show this? As they describe it, their argument is “extremely elementary,” but this only means that it doesn’t use any topological machinery invented since 1970. After all, the question they answer has been on the table since 1982. Elementary can be quite different from easy.