Higher Dimensional Knot Project Description and Ideas/Progress
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Alex
Apr 14, 2008, 1:33:47 PM4/14/08
to knot theory reu
My Project will be working on, and hopefully answering, some questions
about higher dimensional knots. Simple things I want to do include
becoming more proficient at drawing 3-dimensional slices of 2-knots,
and to see if I can find a way to construct 2-knots, perhaps similarly
to how our textbook constructs a 2-knot version of the trefoil. In
addition, I also want to work on invariants of 2-knots. Coloring is
already discussed in our text, and I think it can be generalized to
group labelings fairly easily. More complicated would be attempting to
find a polynomial invariant, or perhaps a more geometric invariant
such as the Vasssiliev invariant for knots in 3-space.
So far, I've read the chapter on higher dimensional knots in our text,
and have chosen several examples/problems I can work that will help me
understand what I need to work towards me "simple" results. Of course,
I've done more interesting things as well.
One idea that I want to work on some more is that of an orientation of
a 2-knot. I have two ideas on that for now, and am leaning towards the
second. The first is that you can orient a 2-knot by simply orienting
the sphere from which the 2-knot is created. When I say orient the
sphere, I imagine something like the magnetic field of the earth,
where you designate one point of the sphere as the top, one as the
bottom, and draw arrows curved from the top to the bottom. The
orientation of a surface, then, might either be some analog of flux,
or some simpler directional idea. This model is a bit troublesome,
though, in that I don't see how it can be useful.
There are some major problems that arise with it, for instance, in
deciding how to measure the orientation not of a point, but of a disk,
and with this model you'll definitely want to orient disks to check
consistency, as the 2-knots intersections with itself are all disks.
One problem that arises this way is that suppose you choose a disk in
the middle of the 2-sphere, stuck between the "top" and "bottom" so
that the "flux" goes straight through the disk. (Another way to think
of this is that, if you choose one side of the disk, say that facing
the bottom, the arrows of flux all point the way this bottom is
facing.) But you can easily deform this disk so that the flux goes
both in and out of one side, not to mention the problems that arise
with you simply being able to stretch the disk to bigger or smaller
proportions. Other problems are the arbitrary picking of a "top" and
"bottom".
The second idea on orientation is probably more workable, but it's
also newer, so I haven't been able to poke holes in it as thoroughly.
The idea is that the 3-dimensional slices of a 2-knot will look like 1-
knots and links (need to check if this is in fact true), and we can
easily orient 1-knots and links. Then, just as colorings of a slice of
a 2-knot follows from the coloring of the slice preceding it, the
orientation will do so as well. A problem arises with giving
orientations to points, but I'm sure simply saying a point is oriented
clockwise or counterclockwise as the disk right above/below it is
oriented ought to work out, but I imagine there might be examples that
ruin this idea. A bigger problem is that I haven't yet checked whether
this method of orienting a 2-knot through orienting its slices even
gives a consistent orientation all the time, as a "band change" might
cause trouble. Perhaps there is some way of addressing band changes,
as the double point resulting halfway through one is what causes the
trouble, and there might be a similar solution to its orientation as
to the orientation of a point.
So orientation is mostly what I'm thinking about right now, in a
theoretical sense. Having said that, I really need some more practice
with slicing knots apart, as I currently have a derth of examples to
at least check some basic ideas on before attempting to prove them.
Still, a little practice should get me proficient fairly quickly.
There are also some trivial results on invariants that generalize to 2-
knots. Naturally, 2-knots have a Crossing Index, defined as it is for
1-knots except that crossings are now disks. The unknotting number
also has a canonical analog. I might do some more detailed thinking on
these invariants later.
That's all for now.
about higher dimensional knots. Simple things I want to do include
becoming more proficient at drawing 3-dimensional slices of 2-knots,
and to see if I can find a way to construct 2-knots, perhaps similarly
to how our textbook constructs a 2-knot version of the trefoil. In
addition, I also want to work on invariants of 2-knots. Coloring is
already discussed in our text, and I think it can be generalized to
group labelings fairly easily. More complicated would be attempting to
find a polynomial invariant, or perhaps a more geometric invariant
such as the Vasssiliev invariant for knots in 3-space.
So far, I've read the chapter on higher dimensional knots in our text,
and have chosen several examples/problems I can work that will help me
understand what I need to work towards me "simple" results. Of course,
I've done more interesting things as well.
One idea that I want to work on some more is that of an orientation of
a 2-knot. I have two ideas on that for now, and am leaning towards the
second. The first is that you can orient a 2-knot by simply orienting
the sphere from which the 2-knot is created. When I say orient the
sphere, I imagine something like the magnetic field of the earth,
where you designate one point of the sphere as the top, one as the
bottom, and draw arrows curved from the top to the bottom. The
orientation of a surface, then, might either be some analog of flux,
or some simpler directional idea. This model is a bit troublesome,
though, in that I don't see how it can be useful.
There are some major problems that arise with it, for instance, in
deciding how to measure the orientation not of a point, but of a disk,
and with this model you'll definitely want to orient disks to check
consistency, as the 2-knots intersections with itself are all disks.
One problem that arises this way is that suppose you choose a disk in
the middle of the 2-sphere, stuck between the "top" and "bottom" so
that the "flux" goes straight through the disk. (Another way to think
of this is that, if you choose one side of the disk, say that facing
the bottom, the arrows of flux all point the way this bottom is
facing.) But you can easily deform this disk so that the flux goes
both in and out of one side, not to mention the problems that arise
with you simply being able to stretch the disk to bigger or smaller
proportions. Other problems are the arbitrary picking of a "top" and
"bottom".
The second idea on orientation is probably more workable, but it's
also newer, so I haven't been able to poke holes in it as thoroughly.
The idea is that the 3-dimensional slices of a 2-knot will look like 1-
knots and links (need to check if this is in fact true), and we can
easily orient 1-knots and links. Then, just as colorings of a slice of
a 2-knot follows from the coloring of the slice preceding it, the
orientation will do so as well. A problem arises with giving
orientations to points, but I'm sure simply saying a point is oriented
clockwise or counterclockwise as the disk right above/below it is
oriented ought to work out, but I imagine there might be examples that
ruin this idea. A bigger problem is that I haven't yet checked whether
this method of orienting a 2-knot through orienting its slices even
gives a consistent orientation all the time, as a "band change" might
cause trouble. Perhaps there is some way of addressing band changes,
as the double point resulting halfway through one is what causes the
trouble, and there might be a similar solution to its orientation as
to the orientation of a point.
So orientation is mostly what I'm thinking about right now, in a
theoretical sense. Having said that, I really need some more practice
with slicing knots apart, as I currently have a derth of examples to
at least check some basic ideas on before attempting to prove them.
Still, a little practice should get me proficient fairly quickly.
There are also some trivial results on invariants that generalize to 2-
knots. Naturally, 2-knots have a Crossing Index, defined as it is for
1-knots except that crossings are now disks. The unknotting number
also has a canonical analog. I might do some more detailed thinking on
these invariants later.
That's all for now.
marg...@math.utah.edu
Apr 15, 2008, 11:10:46 AM4/15/08
to knot theory reu
Hi Alex,
This sounds like a fun project.
As for orientation, I think your second idea matches up with the
notion of surface orientation that we gave in class. We said that an
orientation on a surface with either a choice of clockwise/
counterclockwise at each point in the surface, or a choice of "up".
Using the idea of the cross product, this is the same as choosing x
and y vectors at each point in the surface: travelling from x to y
around the origin is like choosing a clockwise direction, and taking
the cross product of x and y is like choosing an up direction. Now,
if you orient each slice of a knot, you can think of that as pointing
in the x-direction, and you can think of time as pointing in the y-
direction. So that's the connection between your idea and the old
idea.
Can you find any nontrivial knots that are (or are not) equivalent to
their reverse (by which I mean the knot you get by reversing the
orientation)?
You are right that the book discusses coloring of 2-knots. However,
the book does not say that if a knot is colorable, then it is
nontrivial. Take for instance the example of Figure 9.9. This has
the "same" slices as the spun trefoil knot, so I would say that it is
colorable. However, the text indicates that this is a trivial knot!
So how can we really prove that the spun trefoil is nontrivial?
Should we try to understand the Stallings examples? Here is the link:
http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.2307/1970127
Dan
This sounds like a fun project.
As for orientation, I think your second idea matches up with the
notion of surface orientation that we gave in class. We said that an
orientation on a surface with either a choice of clockwise/
counterclockwise at each point in the surface, or a choice of "up".
Using the idea of the cross product, this is the same as choosing x
and y vectors at each point in the surface: travelling from x to y
around the origin is like choosing a clockwise direction, and taking
the cross product of x and y is like choosing an up direction. Now,
if you orient each slice of a knot, you can think of that as pointing
in the x-direction, and you can think of time as pointing in the y-
direction. So that's the connection between your idea and the old
idea.
Can you find any nontrivial knots that are (or are not) equivalent to
their reverse (by which I mean the knot you get by reversing the
orientation)?
You are right that the book discusses coloring of 2-knots. However,
the book does not say that if a knot is colorable, then it is
nontrivial. Take for instance the example of Figure 9.9. This has
the "same" slices as the spun trefoil knot, so I would say that it is
colorable. However, the text indicates that this is a trivial knot!
So how can we really prove that the spun trefoil is nontrivial?
Should we try to understand the Stallings examples? Here is the link:
http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.2307/1970127
Dan
marg...@math.utah.edu
Apr 21, 2008, 12:10:30 PM4/21/08
to knot theory reu
Hi Alex,
Can we draw the films (or make models for) other spun knots? What
about the spun granny knot, or the spun figure 8 knot?
Can we draw the films (or make models for) other spun knots? What
about the spun granny knot, or the spun figure 8 knot?
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