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What Is Knot Theory? Why Is It in Mathematics? [pdf]

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Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#21

Post by catnaroek »

dkarapetyan wrote: post #20
catnaroek wrote: post #19

Earlier quoted context omitted.

Intuition quickly becomes unreliable when you move to spaces with weird topologies, like non-Hausdorff and non-(pseudo)metrizable spaces. When your intuition stops being useful, you actually need to calculate.

I suspect a few folks that studied p-adic numbers extensively would disagree. And in general topologists and algebraists that study non-euclidean things in general.

The p-adic numbers can be equipped with a metric. The induced topology isn't Euclidean, but it's pretty tame compared to what you can see in a general topological space.

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#22

Post by wolfgke »

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants.

But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different simplical complexes or different CW complexes), the dinvariant will assign different values. On the other hand, if the knots are equivalent, the assigned values might not be equal.

What I want to know is: Why doesn't there seem to exist a theory of dinvariants for knots (or topological spaces)?

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#24

Post by pizza »

RachelF wrote: post #6

In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory: https://www.sciencedaily.com/releases/2016/02/160210170411.h... On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.

Loops tangle creating more loops easierly and do not undo themselves typically but rather get tighter upon pulling

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#25

Post by nhatcher »

wolfgke wrote: post #22

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different s…

That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#26

Post by Varinius »

wolfgke wrote: post #22

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different s…

Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different.

Dually, you would want the following: if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.

(full disclosure: lawyer who did his undergraduate degree in math, not a topologist)

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#27

Post by wolfgke »

nhatcher wrote: post #25
wolfgke wrote: post #22

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different s…

That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.

As far as I understand it the far goal is to find invariants that are also dinvariants.

But why don't we build a systematic theory of dinvariants (similar to the theory of invariants) to get a much better understanding of them?

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#28

Post by tgb »

wolfgke wrote: post #22

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different s…

I think it comes down to the difficulty of finding such things. The goal is usually to find "enough" invariants that all objects can be distinguished, but we settle for just whatever invariants we can happen to find. Finding an invariant is pretty easy: it's straight-forward to show that something is invariant under a certain transformation since you just apply a transformation (in this case, usually a homotopy or similar) and see what it can do to the invariant. For knot theory, you can often check just what it does under the Reidemeister moves, giving you just a couple things to check.

How would you go about showing that something is a dinvariant? You would need to take two objects that are distinct and show that their dinvariants are distinct. I.e. use the fact that a homotopy between the two knots does not exist. This is much more difficult, though it does get done. In particular, it's done whenever we have a complete classification of all objects of a certain type (say orientable surfaces, classified by the single invariant (and also dinvariant) their genus). Though generally here we would be actually approaching this from an invariant perspective but just showing that once you have enough invariants, they collectively become a dinvariant.

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#29

Post by wolfgke »

Varinius wrote: post #26
wolfgke wrote: post #22

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different s…

Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different. Dually, you would want the following: if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function,…

> Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different.

> if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.

That doesn't make them useless. Dinvariants just serve a different purpose:

- invariants serve the purpose of distinguishing knots that are different

- dinvariants serve the purpose of detecting that knots that look very different are actually the same

Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]

#30

Post by nhatcher »

wolfgke wrote: post #27
nhatcher wrote: post #25

Earlier quoted context omitted.

That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.

As far as I understand it the far goal is to find invariants that are also dinvariants. But why don't we build a systematic theory of dinvariants (similar to the theory of invariants) to get a much better understanding of them?

Because we can't find them :S. In the case of knots, we would rather have polynomials associated with knots that satisfy your requirement. But we just don't know how to do that. It is the same with topological spaces and homotopy theory or cohomology. Whenever somebody finds a new algebraic invariant that is capable of differentiating between simmilar structures that's a big deal. Like the Jones polynomials did (Or the Donaldson polynomials in the 80's for four manifolds)
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