How Do Knots in 4-Manifolds Influence Homology Classes?

• Bacle
In summary, it seems that the relationship between general knots and homology in the following respect is not clear: it may be that there are no knotted 4-manifolds, but it may also be the case that there are knotted 4-manifolds but their first homology group is trivial.
Bacle
Hi, Everyone:

I am curious as to the relationship between general knots and homology in the
following respect:

Say an orientable surface S is knotted in X^4, an orientable 4-manifold. Then there are

two non-isotopic embeddings e, e' of S in X^4 . Does it follow that H_2(X^4)=/0?

If e(S), e'(S) are both orientable in X^4, then they define respective homology

classes a and b. How can we tell if a~b? I imagine we would use the respective

induced maps e* and e'* , but I don't see where to go from there. I think there

may be an issue of bordism here ( AFAIK, for dimensions n<=4 , homology and

bordism coincide, i.e., if a~b , for a,b in H_k ; k<=4, then a-b bounds a (k+1)-manifold) -- and not

just some subspace. ) Conversely: if I knew that H_2(X^4)=0 . Does it follow that there aren't any S-knots, i.e., that there

is only one embedding of S in X^4 , up to isotopy? . Again, it seems to come down to determining if

we can have a 3-manifold whose boundaries are e(S) and e'(S). I don't see why this could not happen. Any Ideas?

Thanks.

Last edited:
I am confused. Aren't there knotted circles in R^3?

lavinia said:
I am confused. Aren't there knotted circles in R^3?

Yes, you are right. I was wondering if the results generalized to codimension-2
manifolds; R^n is in general an extreme example, since it is too nice in too many ways.

Bacle said:
Yes, you are right. I was wondering if the results generalized to codimension-2
manifolds; R^n is in general an extreme example, since it is too nice in too many ways.

I don't know anything about knots but let's see if we can construct a knotted torus in R^4.

Are you only interested in the second homology group, or is that just the example you chose?

I think it fails for the first homology group, which is the abelianization of the fundamental (first homotopy) group, and therefore is trivial if the fundamental group is perfect. So for example a space with fundamental group A_5 by definition of the fundamental group has non-isotopic embeddings of S^1, but its first homology group is trivial.

At Tinyboss- His example is for 4-manifolds, so the second homology group is the only one that matters to any significant degree.

At OP- As to the second part of your question, If the second homology group is trivial then certainly any two embeddings of S are the boundary of some 3-manifold. However I'm not sure that we can say that these two embeddings are isotopic. In fact I would probably say no. If limit ourselves to a case more familiar, consider S^3. The first homology group of S^3 is trivial, however if we embed two non-equivalent knots in S^3, such as the trefoil and the unknot, they are certainly not isotopic even though the first homology group of S^3 is trivial.

Although this case was for a 3-manifold, I would imagine that you could find a similar example for 4-manifolds.

1. What are general knots?

General knots are mathematical objects that represent the entanglement of a piece of string or rope in three-dimensional space. They are made up of a series of loops and crossings.

2. What is homology in relation to knots?

Homology is a mathematical concept that helps us understand the structure and properties of knots. It is a way of measuring the "holes" or "twists" in a knot by assigning numerical values to its various components.

3. How is homology useful in knot theory?

Homology allows us to distinguish between different types of knots and to classify them based on their properties. It also helps us understand the relationship between knots and other mathematical objects, such as surfaces and groups.

4. What are some applications of knot theory in real life?

Knot theory has applications in a variety of fields, including chemistry, physics, and biology. For example, knot theory can help chemists understand the structure and behavior of complex molecules, and it can aid in the study of DNA and protein folding in biology.

5. Can knots be untangled?

Yes, some knots can be untangled through a process called "ambient isotopy," which involves deforming the knot without breaking or cutting it. However, there are some knots, known as "prime knots," that cannot be untangled using this method and are considered to be "truly knotted."

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