In geometry and the closely related field of topology, adding a spatial dimension can often have wondrous effects: Previously distinct objects become indistinguishable. But a new proof finds there are some objects whose differences are so stark, they can’t be effaced with a little more space.
The work, posted at the end of May, solves a question posed by Charles Livingston in 1982 concerning two-dimensional objects known as Seifert surfaces. Livingston knew that many pairs of certain types of Seifert surfaces are distinct when situated within the tight confines of three-dimensional space, but become equivalent in four-dimensional space where there’s more room to move them around.
“It’s a common thing in topology that when you have something interesting and add an extra dimension, it might not be interesting anymore,” said Maggie Miller of Stanford University and the Clay Mathematics Institute, co-author of the new proof with Kyle Hayden of Columbia University, Seungwon Kim of Seoul National University, JungHwan Park of the Korea Advanced Institute of Science and Technology, and Isaac Sundberg of Bryn Mawr College.
Livingston, now professor emeritus at Indiana University, wondered whether the homogenizing effect of moving into four dimensions applies to all such pairs of Seifert surfaces, or whether there are some that hold on to their uniqueness.
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. It also goes a step further, identifying additional pairs of Seifert surfaces that become alike in some ways but not in others, thereby elucidating the subtle complexities of four-dimensional space.
To understand Seifert surfaces, you need to first understand knots, which you can think of as closed loops of string. The simplest knot is just a circle. More complicated knots include the trefoil and the figure eight, in which the string crosses over and under itself some number of times before its ends join together.
Knots are one-dimensional objects, but they form the boundary of two-dimensional surfaces. Think again about a circular loop of string. That loop forms the boundary of a two-dimensional disc that sits inside it. Similarly, more complicated knots bound more complicated surfaces. A Seifert surface is any surface whose boundary is a knot.
Moreover, a single knot can bound more than one Seifert surface. One way to see how is to picture a knot traced on a surface. It will divide the surface into two complementary Seifert surfaces.
Livingston’s question was about equivalence between Seifert surfaces. Specifically, he was interested in surfaces that can be gradually deformed to look exactly like each other, like a round disc that can be stretched into an oblong disc. Any two surfaces that can be made to perfectly resemble each other in this way are called isotopically equivalent.
Livingston wanted to know whether there are any pairs of Seifert surfaces from the same knot (and of the same genus — meaning they have the same number of holes) that are not isotopic in either three- or four-dimensional space. Over the ensuing decades mathematicians could not find any whose three-dimensional differences survived transport to the higher dimension.
“I put out that question and it sat there for 40 years,” said Livingston.
Miller first became interested in the question last year. This past April, Miller, Hayden and Sundberg were at a conference in Providence, Rhode Island, where they found themselves talking late one evening. Miller mentioned a pair of surfaces she, Kim and Park had in mind for proving Livingston’s conjecture. The surfaces had first been identified in 1975, and while no one had proved that they are isotopic in four dimensions, no one had disproved it either. The surfaces are constructed from a knot with 19 crossings and look like offset images of each other.
To determine whether topological objects like a pair of surfaces are equivalent, mathematicians use a variety of tests. These tests, called invariants, take many forms. Some are more complicated to run and perceive more about the object; others are easier to implement but perceive less.
Hayden and Sundberg had been doing some exciting work recently using an invariant called Khovanov homology, which uses algebra to extract information about how an object is put together. These calculations can be difficult to perform, and for very complicated objects — like a knot with hundreds of crossings — they’re prohibitively hard. But for the knot and associated Seifert surfaces Miller, Kim and Park had in mind, they were very tractable. Determining whether two surfaces are equivalent then becomes a matter of comparing the numbers you get when you calculate each one’s Khovanov homology invariant.
“If you know the numbers are different, you know the surfaces are different,” said Khovanov, who developed his namesake technique in the late 1990s.
Hayden and Sundberg looked at the pair of Seifert surfaces Miller brought to their attention and quickly showed that their Khovanov homology invariants didn’t match, proving the surfaces were distinct.
“It turns out these examples really were hiding in plain sight for a long time,” said Hayden.
At this point the mathematicians had solved Livingston’s question — identifying a pair of Seifert surfaces that are not isotopic in three-dimensional space and remain that way in four — but they didn’t stop there. Emboldened by the news from Hayden and Sundberg, the group decided to see if they could prove the same result using more basic invariants. They did, using a fundamental technique in topology called a branched covering.
“They answer a question that’s 40 years old with about a page of mathematics,” said Patrick Naylor of Princeton University. “That doesn’t happen very often.”
They also went on to identify additional pairs of Seifert surfaces that, in four-dimensional space, are not isotopic in one sense (they’re not isotopic “smoothly”) but are in another (they’re isotopic “topologically”). This distinction, between topological and smooth equivalence, doesn’t apply in three dimensions and is one of the deep mysteries of four-dimensional space — though less so now for Seifert surfaces.