Linked Homotopy Theory: Preserving Linking Number Zero

[wofsy]

Is there a theory of linked homotopy? I am thinking of homotopies of 3 space minus some other loops - where the loop is not allow to intersect itself during the homotopy.

This type of homotopy would preserve linking of 2 loops that have linking number zero.

 

[owlpride]

am thinking of homotopies of 3 space minus some other loops - where the loop is not allow to intersect itself during the homotopy.

Isn't that exactly what you get in standard knot theory when you add another knot where the "removed" curve in R^3 would go? A good example in this area are the Borromean rings: it is a three-component link in which any two components have linking number zero, but all three together cannot be taken apart because the third component obstructs the ambient isotopy.

 

[wofsy]

Isn't that exactly what you get in standard knot theory when you add another knot where the "removed" curve in R^3 would go? A good example in this area are the Borromean rings: it is a three-component link in which any two components have linking number zero, but all three together cannot be taken apart because the third component obstructs the ambient isotopy.

Suppose I want a homotopy of the Borrmean rings where two of the loops are moved to lines perpendicular to the xy-plane and the third one is carried along without ever intersecting either of the other 2? (should it be allowed to cross itsef?)More generally can I use homotopies like this to come up with an idea of equivalence?

Another odea of equivalence might be to just compute the homotopy class of the third loop in the fundamental group of S^3 - other 2 but I would like to do it with homology rather than homotopy - if possible.

 

[owlpride]

In standard knot theory, if you move two of the Borromean rings, the third one is automatically carried along. Whether or not the the individual components are allowed to intersect themselves in the process depends on whether you are looking at link homotopies or link isotopies.

All the theories I am aware of assume symmetric conditions though: either no knots may intersect themselves or all knots may intersect themselves. Not sure what happens if one of the loops is given a special status. As you have noted yourself, it amounts to doing knot theory in a nontrivial 3-manifold.

 

[wofsy]

In standard knot theory, if you move two of the Borromean rings, the third one is automatically carried along. And no, the third knot would not be allowed to intersect itself during the isotopy. Knot theory never allows knots to intersect themselves.

If you do no allow self intersection then it seems that you will preserve linking where thw linking number is zero. Suppose I want an equivalence by linking number.

I am not sure though if knot theory is what you are looking for. In your example, you say that you want two of the borromean rings to be carried to a line perpendicular to the xy-plane. Knot theory would not allow you to do that because a circle mapped to a line necessarily self-intersects.

A meant two parallel lines. It is tru that you would get intersection at the north pole of the 3 sphere but this would not create new linking and the Borromean link pattern would preserved.

If you want to look at standard loop homotopies (allowing self-intersections) in R^3 minus a loop (which is not allowed to intersects itself), you are computing fundamental groups of 3-manifolds. 3-manifolds have not been completely classified yet, but I would assume that their fundamental groups have been studied quite a bit?

It seems that standard loop homtopy describes the Borromean rings since the third loop forms a commutatos in the free group on two generators - I think

 

[owlpride]

I am not sure if that's what you are looking for, but there are "higher dimensional" linking numbers due to Milnor. In the case of the Borromean rings, the 3-dimensional linking number would tell you that the three rings are linked, although their pairwise linking numbers are zero. These higher-dimensional linking numbers are called mu-bar or Milnor invariants. Massay products (a cohomology construction) capture the same idea.

That's about all I know about this topic. Maybe someone else can better answer your question.

 

[wofsy]

I am not sure if that's what you are looking for, but there are "higher dimensional" linking numbers due to Milnor. In the case of the Borromean links, the 3-dimensional linking number would tell you that the three rings are linked, although their pairwise linking numbers are zero. These higher-dimensional linking numbers are called mu-bar or Milnor invariants. Massay products (a cohomology construction) capture the same idea.

That's about all I know about this topic. Maybe someone else can better answer your question.

Thanks for your response. I appreciate it.

I have been trying to figure this problem out on my own - like a big homework problem. The homotopy problem seems too technical for me to solve.

My approach was to find a standard form using some kind of homotopy for links then look at the cross products of the magnetic fields that the loops produce when currents are passed through them. I still think this will work.

For instance you know that if 2 loops are unlinked then the cross product of the duals(using the Euclidean metric) of the magnetic fields is an exact differential 2 form. I am having trouble picturing this form.