What Are Knot Trees?
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onye
Apr 21, 2008, 1:08:36 PM4/21/08
to knot theory reu
Another way of finding the determinant is using trees, which are
special kinds of graphs. A graph consists of a set of points called
vertices and a set of edges that connect them. A tree is a graph that
is connected and has no closed cycles.
From a projection of an alternating knot or link, we create a
corresponding planar graph in the following way:
1. Shade the diagram in a checkerboard pattern.
2. Put a vertex at the center of each shaded region.
3. Connect the vertices with edges that pass through a crossing.
Note: the determinant of an alternating knot is the number of maximal
trees in its planar graph.
http://www.math.jmu.edu/~taal/OJUPKT/lopez.pdf
special kinds of graphs. A graph consists of a set of points called
vertices and a set of edges that connect them. A tree is a graph that
is connected and has no closed cycles.
From a projection of an alternating knot or link, we create a
corresponding planar graph in the following way:
1. Shade the diagram in a checkerboard pattern.
2. Put a vertex at the center of each shaded region.
3. Connect the vertices with edges that pass through a crossing.
Note: the determinant of an alternating knot is the number of maximal
trees in its planar graph.
http://www.math.jmu.edu/~taal/OJUPKT/lopez.pdf
marg...@math.utah.edu
Apr 21, 2008, 1:13:11 PM4/21/08
to knot theory reu
Hi Onye,
Sounds good. What is a maximal tree? Is this easier than computing
things the old way? Can we obtain the formula pq+pr+qr from this?
Any ideas on how to prove this theorem?
Dan
Sounds good. What is a maximal tree? Is this easier than computing
things the old way? Can we obtain the formula pq+pr+qr from this?
Any ideas on how to prove this theorem?
Dan
onye
Apr 21, 2008, 1:32:08 PM4/21/08
to knot theory reu
Maximal Trees are simply the maximum possible number of tree diagrams
you can construct from a knot diagram. I think the procedure is much
simpler than the determinant method in that you don't have to worry
about complex computations for knot diagrams with many crossings. All
you've got to do is just count the number of trees obtainable from the
diagram. I think this concept is pretty new and still evolving.
Nevertheless, it's a beautiful concept, very similar to the activity
we undertake when trying to determine the number of Seifert surfaces
on a knot diagram.
> >http://www.math.jmu.edu/~taal/OJUPKT/lopez.pdf- Hide quoted text -
>
> - Show quoted text -
you can construct from a knot diagram. I think the procedure is much
simpler than the determinant method in that you don't have to worry
about complex computations for knot diagrams with many crossings. All
you've got to do is just count the number of trees obtainable from the
diagram. I think this concept is pretty new and still evolving.
Nevertheless, it's a beautiful concept, very similar to the activity
we undertake when trying to determine the number of Seifert surfaces
on a knot diagram.
>
> - Show quoted text -
Jason
Apr 25, 2008, 3:07:36 PM4/25/08
to knot theory reu
This is an interesting concept. in arc presentations of a knot they
talk about something called the spanning tree of a knot and i was
wondering if there is any connection between these two "trees". a
spanning tree is were you take a diagram of a knot and you collapse
all the crossings to give you a curve that is no longer simple. Then
you more or less cut the arcs so that you have no crossings that are
connected. I will post a picture in files.
> > >http://www.math.jmu.edu/~taal/OJUPKT/lopez.pdf-Hide quoted text -
talk about something called the spanning tree of a knot and i was
wondering if there is any connection between these two "trees". a
spanning tree is were you take a diagram of a knot and you collapse
all the crossings to give you a curve that is no longer simple. Then
you more or less cut the arcs so that you have no crossings that are
connected. I will post a picture in files.
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