# How does non-isotopic embeddings affect the homology of manifolds and knots?

• Bacle
In summary, the conversation is discussing the relationship between non-isotopic embeddings of a subspace N in a manifold M and the n-th homology of M, where n is the dimension of M. It is mentioned that if there are non-isotopic embeddings of N in M, then M is knotted or N is a knot in M. The conversation also considers the orientability of N and the inclusion of N into the space, as well as the determination of the homology class of the induced map for non-isotopic embeddings. There is a question about whether H_n(M) is trivial if there are no knots, and if H_n(M) is not trivial if there are non-isotopic embeddings of N in
Bacle
Hi, everyone:

I think this should be simple, but I've been stuck for a while now:

Let M be an n-manifold and N an n-dim. subspace of M, of possibly lower dimension

than M , then M is knotted, or N is a knot in M if there are non-isotopic

embeddings of N in M.

Now:

Say there are non-isotopic embeddings of N in M. How does this affect the

n-th homology of M (n is the dimension of M) ? . Does it follow that if

H_n(M)==0 , that there are no knots, i.e., there is only one isotopy class

of embeddings of N in M? And, conversely, if f,g, are two non-isotopic embeddings

of N in M, does it follow that H_n(M) is not trivial?

Thanks .

This is not a full answer by far, but I think we may be able to use the

map induced on homology by the inclusion of the subspace into the space.

I believe --please correct me if I am wrong -- that if the subspace N is

orientable , and the included image i(N) is a submanifold, then i(N) may be/represent

a homology class. And then the issue seems to be that of determining the

homology class of i*(N) , i'*(N) , for non-isotopic embeddings i, i'.

## 1. What are general knots and homology?

General knots are mathematical objects that can be described as a closed loop or a tangled arrangement of a single piece of string. Homology, on the other hand, is a mathematical concept that studies the properties of these knots, specifically their invariants or characteristics that do not change under certain transformations.

## 2. How are general knots and homology related?

Homology is used to classify general knots by assigning them a numerical value or "invariant" based on their geometric properties. This allows for the comparison and distinction of different knots, even if they have the same shape or appearance.

## 3. What is the significance of studying general knots and homology?

The study of general knots and homology has many real-world applications, including in physics, chemistry, biology, and computer science. It also has implications in topology and geometry, providing a deeper understanding of the fundamental structures of our universe.

## 4. What are the main techniques used to study general knots and homology?

The two main techniques used to study general knots and homology are algebraic topology and geometric topology. Algebraic topology uses algebraic methods to analyze the properties of knots, while geometric topology focuses on the geometric aspects of knots.

## 5. What are some open questions in the field of general knots and homology?

Despite decades of research, there are still many open questions in the field of general knots and homology. Some of these include the classification of all possible knots, the computation of invariants for more complex knots, and the development of more efficient algorithms for knot analysis.

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