Adding a spatial dimension can make previously distinct objects indistinguishable. There are some objects that can't be compared with a little more space.
The work was posted at the end of May and solved a question posed in 1982. In four-dimensional space, where there is more room to move them around, there are many pairs of different types of Seifert surfaces that are 888-609- 888-609- 888-609- 888-609- 888-609-
The co-author of the new proof said that when you have something interesting and add an extradimension, it might not be interesting anymore.
Livingston wondered if the effect of moving into four dimensions applied to all pairs of Seifert surfaces, or if there were some that held on to their unique characteristics.
The first pair of Seifert surfaces that are clearly different from each other in four dimensions are identified by the new work. Additional pairs of Seifert surfaces that become alike in some ways but not in others are identified.
The first thing you need to understand is knots, which are closed loops of string. There is a simple knot. The trefoil and the figure eight are examples of more complicated knots.
The boundaries of two-dimensional surfaces are formed by knots. Think about a loop of strings. There is a two-dimensional disc in that loop. More complicated knots bind more complex surfaces. Any surface with a knot is called a Seifert surface.
A single knot can bind multiple surfaces. A knot is traced on a surface. The surface will be divided into two parts.
Livingston wanted to know if there was any equivalency between the two surfaces. He was interested in surfaces that can be gradually altered to look like each other, like a round disc that can be stretched into an oblong disc Any two surfaces that can be made to look exactly like each other are called isotopically equivalent.
Livingston wanted to know if there were any pairs of surfaces from the same knot that did not have the same number of holes. The three-dimensional differences that mathematicians were unable to find over the ensuing decades.
It sat there for 40 years after Livingston asked it.
Miller was interested in the question a year ago. They were at a conference in Providence, Rhode Island, where they found themselves talking late one night. Miller mentioned a couple of surfaces that Kim and Park had in mind. No one had disproved the idea that the surfaces are in four dimensions. The surfaces are made from a knot and look like pictures of each other.
A number of tests are used to determine if a pair of surfaces are the same. Many forms of the invariants tests are taken. Some are more difficult to run and see than others.
Some exciting work was done recently using an invariant called Khovanov homology, which is used to extract information about how an object is put together. For very complicated objects, like a knot with hundreds of crossing, they are difficult to calculate. They were very tractable for the knot and associated surfaces. It is important to compare the numbers you get when you calculate each one's Khovanov invariant to determine whether or not two surfaces are equivalent.
If you know the numbers are different, you can tell the difference between the surfaces.
The surfaces Miller brought to their attention were different from the other ones.
The examples were hidden for a long time.
The mathematicians had solved Livingston's question by identifying a pair of Seifert surfaces that were not isotopic in three-dimensional space and remained that way in four, but they didn't stop there. The group decided to use more basic invariants in order to prove the same result. The technique they used was called a branched covering.
The answer to the question is 40 years old with about a page of mathematics. It doesn't happen a lot.
There are more pairs of Seifert surfaces that are not isotopic in one sense and are in another. One of the mysteries of four-dimensional space is the fact that the distinction between smooth and topological doesn't apply in three dimensions.
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