View Full Version : Knot Polynomials

Sir Knots A Lot

2010-Aug-15, 04:33 PM

Is anyone here familiar with the math of knot polynomials?

I'd like to know about methods for using these equations in combination with other linear equations.

grapes

2010-Aug-15, 06:03 PM

Is anyone here familiar with the math of knot polynomials?Only for homework :)

I'd like to know about methods for using these equations in combination with other linear equations.There are different types of knot polynomials, and they're not linear nor equations. How are you planning on using them?

Sir Knots A Lot

2010-Aug-15, 07:46 PM

Well, I'd like to know if the dirac equation can be generalized for a trefoil topology.

Sir Knots A Lot

2010-Aug-16, 01:56 PM

So yeah... them knot polynomials and Dirac's equation. No dice on that one, eh?

Sir Knots A Lot

2010-Aug-17, 03:17 PM

Well, could someone explain knot polynomials to me a little. I'd just like to understand what can be done with them.

Specifically the trefoil knot.

grapes

2010-Aug-18, 10:40 PM

Specifically the trefoil knot.There are wiki pages that describe a lot of the mechanics of knot polynomials (http://en.wikipedia.org/wiki/Knot_theory). The Alexander polynomial of the unknot (http://en.wikipedia.org/wiki/Unknot) is 1, and since the polynomial is an invariant of the knot, any knot that doesn't have a polynomial of 1 is not an unknot. The polynomial of the trefoil is 1 + z2, so it is not an unknot. However, both left- and right-handed trefoils have polynomial 1 + z2. THere are some knots that have polynomial 1 that are not unknots.

However, there are more than one way to compute a knot polynomial. The Jones polynomial can tell the difference between the two types of trefoil knots, and it is apparently an open question whether any knot shares the same Jones polynomial as the unknot.

Sir Knots A Lot

2010-Aug-19, 02:21 AM

Is there a way to examine the mathematics of the crossing themselves?

For example, if one were to expand one loop of the knot, the remainder of the knot would tighten and produce 4 unique corners.

Is there a way to mathematically express knot characteristics in this manner?

grapes

2010-Aug-19, 03:46 AM

Is there a way to examine the mathematics of the crossing themselves?I'm not sure what you mean by that. The knot polynomials take that into account, in a topological way.

For example, if one were to expand one loop of the knot, the remainder of the knot would tighten and produce 4 unique corners.

Is there a way to mathematically express knot characteristics in this manner?So, you're just tightening up the crossings? From a knot/topology perspective, that doesn't really change things.

Besides, why do you call them "unique"? What is it that makes them unique, to you? Defining that might be a place to start.

Sir Knots A Lot

2010-Aug-19, 04:02 PM

Hmmm... perhaps it would be better to say each of the corners is 'unique' from each other.

Two crossing would close against the loop thats being expanded and the other two would close against each other.

As an example, if it were possible to tie field lines into a knot, what would be the strength of the field at each of the corners?

grapes

2010-Aug-19, 05:21 PM

Hmmm... perhaps it would be better to say each of the corners is 'unique' from each other. Then it would be sufficient to say "four corners" rather than "four unique corners". They might be four identical (not unique in some sense) corners.

Two crossing would close against the loop thats being expanded and the other two would close against each other.

As an example, if it were possible to tie field lines into a knot, what would be the strength of the field at each of the corners?Field lines are mathematical contructs not necessarily physical lines. The strength of the field depends upon the kind of field--and some fields might be impossible to knot. What field do you have in mind?

Sir Knots A Lot

2010-Aug-19, 05:32 PM

Then it would be sufficient to say "four corners" rather than "four unique corners". They might be four identical (not unique in some sense) corners.

Field lines are mathematical contructs not necessarily physical lines. The strength of the field depends upon the kind of field--and some fields might be impossible to knot. What field do you have in mind?

I'm actually examining the Dirac equation.

It's difficult to describe what I'm trying to accomplish. I noticed a similarity between quark structure and the trefoil knot, provided one loop is expanded. I've extended the thinking to cover neutrons, which allows modelling of higher nuclear structures based on a sort of principle of least action.

But the modelling is very simple, just showing how down quarks and up quarks would arrange themselves within the nucleus to minimize the appearance of fractional charge.

But it would be nice to be able to mathmatically express what I'm trying to model.

grapes

2010-Aug-19, 07:50 PM

I noticed a similarity between quark structure and the trefoil knot, provided one loop is expanded. What is that similarity?

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