What is Topology, and what's the big deal?

The diagrams are supposed to be planar projections of part of a loop in three dimensions. Think about how many ways a planar crossing can appear in/as the shadow of a loop in three dimensions.Note the diagrams aren't closed because they represent parts which are connected to a larger diagram which is closed.

I use the idea of being able to form a stable surface, like a soap film, somewhere along the boundary; recall that a knot is a closed loop embedded in three dimensional space, so it's a boundary de facto.

If this boundary is shaped such that a stable 'soap film surface' can form somewhere, that's obviously a subset of all the embeddings. You can link two or more loops together in three dimensions (the loops are called components), so the number of components linked together is another set of subsets. And, soap films are surfaces which are in physical equilibrium, much like a power cable strung between upright poles; the surface tension is the same everywhere.
Seriously, google soap film physics, you should get some video lectures hosted by universities.

Kauffman doesn't really discuss surfaces, but does cover checkerboarding (so-called), which is a coloring problem for planar graphs. He also tends to start with the trefoil or Hopf link (a 2-component link) a lot; I've been going the other way with components that have two or one crossing each, no crossings is the unknot and so working back up, what can we distinguish? As Kauffman says it's about distinction . . .

See, for me, if I go to the trouble of reading a book by someone I can't help but try getting into their mode of thought (as in, where is this all going, why does he think that detail is important, etc).

 

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Um, No. What that picture looks a lot like, is a planar graph embedded on the torus; the graph is "moving" (rotating or winding around on the surface) but not "changing" since it stays connected. Duh.
Can you tell us what it has to do with knots or say, circles?

It's not really difficult to form a wire boundary of a Mobius strip, or a profile of Homer Simpson.

 

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I can, but duh, I won't. Because it's quite clear you haven't looked at what I said previously, and that you don't have a clue about all this.

Yeah whatever.

Ignore.

 

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Is that a retraction, or is it the sound of someone's soap bubble bursting?

 

This distinction between a figure 8 and a "folded" figure 8 (that, is, folded at the crossing point) in terms of a minimal surface is demonstrated quite well by this video:



Watch what happens to the surface, and where holes appear. The video's maker wanted to show that breaking the film depends on (a point of) contact between the edges.
What I wanted to see was what happens to the surface since it "wants" to minimise itself, as the boundary folds and unfolds.

You can see that there are two distinct ways the loop is being manipulated so the soap film surface changes: by twisting and by folding together, and untwisting and unfolding.

 

Interesting video and clearly illustrates why I asked my question and noted that just a 3D loop folded to touch it self change topographically from one to doubly connected domain when it self touches.

Normally if one wants to make 2D projections of a 3D object at least three orthogonal views are used. (top, side and end views). If for example there are only minor difference between say the two end view you can superpose the differing details of one end on the given view with short dash lines to avoid a separate drawing of it.

I gather from your post you are interested in the 3D object, a doubly connected closed loop, which when parts of it "self touch" becomes more multi path connected than the originally double connected "doughnut."

I am reasonably familiar with nature's skill as a mathematician in solving surface energy minimizing problems - I. e. making soap films on a complex wire structure. - What I could not do in days of analysis, for even some, relative simple, regular, 3D "stick figure" She does in an instant!

 

The self-touching part is what changes the surface (part of it breaks away), what I'm interested in is the mathematical description of both the forms the surface has.

Kauffman introduces his bracket formalism straight away in his book, it's fairly straightforward until you get to where he "normalises" a bracket. He doesn't explain what the regions A or B are supposed to represent but does explain how to locate them, but to me they are like functions that identify the type of switch, they're just A and B type switches which are on. In fact an A type switch is a B type switch rotated 90 degrees, with "inputs and outputs" reassigned, or put another way, given a crossbar switch which is on, it can be used in one of two ways to connect a pair of tracks.

In switching theory of course there aren't any minimal surfaces between tracks, and closing a track by connecting its ends together probably isn't useful, but there is a connection otherwise: consider switching a type A for a type B somewhere, this might undo a knot. If all the switches are on then switching the switches means rotating them all and reassigning inputs/outputs.

 

On that theme of part of the surface breaking, this happens several times in the video. The author breaks an enclosed part with part of the loop thingy he's using. This looks like something springy that's been wrapped with cloth so it absorbs a lot of soapy water.

The other way a part of the surface 'breaks away' in the video demonstration, is when the equivalent of a type I move is made-- a curl is uncurled and part of the surface disappears, its from 27 to 33 seconds; it looks like he uses the tension in the springy thing he's holding to cancel the tension in part of the surface and it collapses--the boundary of the soap film reconfigures under its own tension.

There is one other way he could 'break' the surface, blow on it. What breaks away is a spherical surface, a bubble. It's a sphere because that's the shape of an unbounded minimal surface in the ambient space. In reverse, a soap film is a sphere embedded in a different way, because it can have one or two sides when its a surface that does have a boundary.

 

So I'm thinking, Kauffman wants us to consider the loop in the video that 'contains' a soap film is a figure, a symbol, in three dimensions.

Take some projection of this figure (i.e. stop the video and draw the figure with crossings represented by small breaks in an undercrossing segment), then consider the crossing(s) in a local sense, with small regions around them. Use a version of the right hand rule as follows: if you point your right thumb towards the crossing along an undercrossing segment with the fingers extended, the fingers point to the region on the left of the segment; label this region A. There should be two A regions because there are two undercrossing segments; likewise there are two B regions, the A and B regions are disposed in a 'square' fashion around the crossing.

You should find there is only one crossing in a 'good' projection, but the minimal surface can be a type A or B in the neighbourhood of the crossing. Recall that in terms of a 2x2 crossbar switch which stays closed (i.e. crossed over), A and B are rotations of each other plus appropriate assignments of input (segments) and output (segments).

However, does the existence of an actual surface, near a crossing, change anything? In Kauffman's formalism it doesn't matter if the figure is twisted or folded, what matters is if it's knotted, and the figure in the video isn't knotted; it's the unknot.

 

A digression. Making nice spherical soap bubbles is child's play at atmospheric pressure.

Does the shape of the bubbles depend on them having a slightly higher internal air pressure?
What if there is no atmosphere? Can soap bubbles be made in the conventional way in a vacuum where there is no internal or external air pressure, only surface tension and hydrogen bonding, and of course a loop of some kind of physical material to immerse in a soapy solution? Let's allow gravity to make it easier, so say we're on the surface of the moon or that there's a large enough vacuum chamber handy in which to experiment with soap and potential bubbles . . .

Ok?

I think the first question is easy to answer, dip the end of a drinking straw into some detergent, then blow a bubble. Stop blowing and block the free end of the straw with a finger, hold the straw + bubble in front of you and remove your finger. What happens and why?

 

Yeah, well.

When a soap film forms "between" adjacent parts of a loop boundary, it isn't related to the presence or absence of an atmosphere. What you would need to do to investigate if bubbles can form in a vacuum is accelerate the loop. Maybe have a high speed camera as well, because any bubbles shouldn't last long.

Back to the mathematics. Make a pair of discs out of paper, one smaller than the other and tape a reasonably long strip of paper to them so they're connected along part of their boundary. Ok so now there is a continuous path from a point on one disc to the other; the whole structure has two sides, and there is no continuous path from one side to the other (going "over the edge" isn't allowed).

You can prove it has two sides by coloring each side (so they can be distinguished). Then any twists in the connecting strip will also have two sides. This structure is a mathematical model as well, of the soap film in the video when it's twisted and untwisted.
To model the soap film when the surface changes, place the smaller paper disc inside the larger. Now the paper strip connects the internal disc to the outside of the larger disc, so it can't be a minimal surface (!). The inner disc + strip must be the boundary of a surface but not contain one.

In either configuration, side by side or one inside the other, the twists in the paper strip are either left or right handed (according to the modified right hand rule as above). A left handed twist is the inverse of a right handed twist, and, viewed as a 2x2 crossbar switch a left handed switch is a right handed switch with a "change of indices".

Knots as boundaries have a natural mathematical structure; Kauffman develops tensor diagrams (aka Penrose), and shows that knots, given a matrix-type structure are also abstract tensors. So if you did make a paper model and color it, you have the beginnings of the tensor space over knots and their planar diagrams!

 

So far, so vague.

And there's a thing: it's actually better to think vaguely about mathematical structures and what you can do with them, in terms of modeling their 'structure'. Here, I'm going to conjecture that it would be challenging for any student of topology to banish the notion of shape, since it's a really fundamental thing we distinguish. Geddit?

However, it may have been mentioned that sets don't have this thing called a shape; sets don't really have any structure either until one is defined on them. Sets is where it all starts, hence one might consider that topology is not about shape, and, it isn't in the sense it's what's left after all the shape has been "modded out".

You could write a computer program that computes (the structure of) minimal surfaces, or just make soap films that do the same thing (except you don't then have the 'information' in the same sense a computer memory does).

You could also write a program that deals with a more 'compact' algebraic model, like the 2-colored paper one, and certain operations on its edges (you get the program to remove the interiors and just compute with the boundary, in a representative form, for instance the paper discs in adjacent or concentric disposition.
And the program will have to 'handle' twists in the connecting strip; note that a single twist in the paper strip means one of the discs has been flipped 180 deg. and has a different color. 360 deg. of twist means both discs are the same color; there is a well defined function in terms of odd and even multiples of \( \pi \), this function 'encodes' twist.

The elements A and B are then left/right handed directions for twist. Clearly A and B are commuting variables, since AB = BA = no twist.

Moreover, the relation between an A-type and a B-type crossing can be represented as a function that changes pairs of inputs/outputs in an abstract switch which is on; turning the switch off changes the 'embedding'. This can all be done with sets of distinguished inputs and outputs in pairs, although 'input' and 'output' are more abstractions, of the ends of some connection.

 

A child will tell you that if you have a circular wire loop, it will make soap bubbles when you dip it in soapy water. If you ask them what happens when you twist the loop like a figure 8 so the loop "touches itself" or nearly so, they should say it will make two soap bubbles at a time, when you hold it in a decent breeze.

What, if anything, can this distinction tell us about anything physical? the appearance of two distinct surfaces is indeed apparent if the loop isn't actually touching and the (minimal) surface is continuous and connected. Is there a way to describe it mathematically? When the surface is two sided as it is for a simple closed loop, twisting it leaves it with two distinct sides, as the paper disc model tells us.

When you bring the two parts of a twisted loop together (folding it at the crossing), the surface reconfigures and becomes one sided like a Mobius strip. It isn't a "good" bubble maker any more because now it fits on a torus, in fact the boundary corresponds to a torus knot which winds twice around the axis of rotational symmetry, and once around the interior circle.

Farsight has actually contributed something here, although the graph on a torus doesn't look like a knot it is winding around both of the above symmetry lines 'inside' a torus. My calling it a planar graph isn't correct either, a planar graph embeds in the plane with no edges crossing. However, small regions of it look like an hexagonal lattice so there aren't any smooth curves wound around the torus, so . . .

Here are some pretty pictures (from a Wikipedia link) of the (2,3) torus knot aka the trefoil, which demonstrate sans equations that it's equivalent to the (3,2) torus knot.
Work backwards through (2,2) and (2,1) and there we are.

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Kauffman's book is definitely a challenging read, on the other hand it seems to contain quite prosaic examples (for instance he explains that you can map the quaternions to movements of one of your arms), the Belt trick and so on are in there. Perhaps he's trying to say that all the complicated looking math is really based on quite simple ideas, ultimately.

So the "difficulty" in understanding what the theories are saying, which means having to wade through pages of equations of abstract symbols, is in some sense a failure of imagination.

There, I said it.

 

There aren't any Tex characters you can post for knot math, so I'm wingin it.

Just applying a right hand rule to crossings gives you a way to "generate" the two types, if you include the rule that a pair of lines (arcs) in the plane can be disposed horizontally or vertically, i.e. they have 90 degrees 'between' them, a distinction.

This disposition thing can be reasonably general, in that you can consider a pair of vertical arcs are a pair of semicircles (with vertical diameters), or approximately so, they might be drawn like this say: \( \supset\subset\) , likewise with horizontally disposed arcs.

So, you have a way of labeling, with an A or a B, a pair of horizontal or vertical arcs that describes the actual crossing (or switch) exactly. A particular crossing is really the union of the two ways to describe it (A or B 'acting' on a horizontal or vertical pair of arcs); you want \( A_{=}\;+\; B_{\Vert} \), or \( B_{=}\;+\; A_{\Vert} \), which is exactly like a vector product.

If you multiply \( \begin{pmatrix} A\\ B \end{pmatrix}^T\) or \( \begin{pmatrix} B\\ A \end{pmatrix}^T\) by \( \begin{pmatrix} =\\ \Vert \end{pmatrix}\), the results are either a type A or a type B crossing, as the union of 'states' each type of crossing is in. That is, an A type 'switch' which is always on, is the union of states: {A acting on a pair of horizontal arcs} , {B acting on a pair of vertical arcs}; to generate a B type, exchange the rows of the first matrix as above.

Then any knot or link has this description of each crossing as the union of A and B acting on horizontal or vertical pairs of arcs. A knot state is a kind of decomposition of a knot into these A and B states (of each crossing).

The polynomial is a sum over all the states of a knot. This would be a lot easier to show if I could post knot symbols.
Exercise: the trefoil has 3 crossings and decomposes into \(2^3\) states (convince yourself this is true using Kauffman's notation). If a knot has n crossings, does it have \(2^n\) states?

 

The column vector with just the "pairs of lines" in it should have angle brackets around each symbol, I tried to do this with Tex and failed.

The angle brackets appear in the formal description of a knot or link polynomial thusly:

\( \langle K \rangle \;= \;\langle K \rangle (A,B,d)\;=\; \sum_{\sigma} \langle K \vert \sigma \rangle d^{||\sigma||} \)

The d is interpreted as an operation that does: \(\langle \bigcirc \cup K \rangle \), it "adds" a disjoint circle to a knot diagram, so that \( d\langle K \rangle\;=\; \langle \bigcirc K \rangle \) (it's not a problem omitting the set union symbol).

The \( ||\sigma|| \) is interpreted as the number of disjoint circles less one, in a diagram.​

 

So it seems that we have two quite distinct kinds of objects so far: crossings (in planar knot diagrams), and disjoint circles. The first object is a local feature, the second is a global feature of a particular knot state.
The term \(d^{||\sigma||} \) is d to the power of one less than the number of disjoint circles, or equivalently Jordan curves, in a knot state diagram. In a sense it means A and B "mix" with d, and d has a solution in terms of A and B (you want d to commute with A and B).

The decomposition of knot diagrams into abstract states is a global kind of formulation, the trefoil has a state which is equivalent to three disjoint circles connected by three B crossings. Clearly the state diagrams include local and global information, the previous state is succinctly represented by \( B^3\langle \bigcirc \bigcirc \bigcirc \rangle = B^3 d^2 \langle \bigcirc \rangle \). The three B connections can go anywhere as long as they connect three disjoint circles.

There is a "dual" state with three A crossings: \( A^3 \langle \circledcirc \rangle\) with two disjoint circles.

 

So what is the big deal?

I've banged on about knots in a way that's more "1001 things you can do with knots and their planar projections"; Kauffman gets serious about what knot topology can tell us about the structure of spacetime, where the Lorentz group "lives", chaos theory, Penrose spin networks . . .

Knot theory encompasses low dimensional topological spaces--a knot is homeomorphic to the boundary of a disc in \( \mathbb R^2 \), it's one dimensional. Topology is a vast subject on the other hand.

But maybe the connections between knots and physics is just another way nature is being conservative, with the kinds of symmetry we find in the universe. Mathematics has a vast number of symmetry groups, we don't really see more than a few in nature. Knot theory connects two and three spatial dimensions together, in some sense.

But, hey, if you've ever had to use a garden hose which has been left in a coil on the ground, you're familiar with Reidemeister moves in the plane (given that the ground is relatively flat or smooth); just sayin'.

 

One more obvious thing about knot diagrams and Kauffman's bracket notation:

There is only one way to connect three disjoint circles together so you get a trefoil and not its mirror image; to realise the mirror image, replace A with B and B with A (i.e. switch the switches).

Likewise there is only one way to connect two disjoint circles (oops, actually there are two, one uses all A type crossings as in \( A^3d\)). With just one circle you have to be careful about the fact that A and B are 90 degrees apart. Since there are three crossings in a trefoil, there are 3 ways to 'label' a single circle, 3 ways to label two circles (that's 6 states), and the remaining 2 have been accounted for, that's a total of 8 ways to 'construct' a trefoil; and there are the corresponding 8 ways to construct the mirror image.

You can represent a trefoil bracket state with one, two, or three circles plus connections between boundaries. Playing around with different combinations to see if they represent a knot (or not) might be some assistance in getting one's head around the algebra. For instance, AAB as a braid gives a trefoil knot when you close it (therefore, BAA does too), but ABA doesn't. Constructing a trefoil this way also demonstrates that you first construct a (2,1) torus knot or a figure 8, then a 2-component link, then the knot (assuming one crossing at a time).

Getting to the knot diagrams as abstract tensors is a little more abstract. For example, Kauffman draws a single strand (vertical arc), labels the ends and pronounces it equivalent to the Kronecker delta . . .

Yeesh.

 

Maybe I can understand the abstraction with something a little easier than a labeled strand.

The Kronecker delta is related to the identity matrix in the following way: the indices of \( I_n \) denote the row and column for each element in the nxn matrix, and each element of \( I_n \) is 0 or 1. Therefore \( \delta_{ij} \) "determines" the entries in \( I_n \).

But we can't say \( I_n = \delta_{ij}\), though we can say \( (I_n)_{ij} = \delta_{ij}\). The indicies appear on both sides, and the latter equation says that element ij of the identity is equal to the Kronecker delta, which is not a matrix.

The Kronecker delta can be considered as a function that inputs two numbers and outputs a third number, 0 or 1. We also have that \( \delta_{ij} = \delta^i_j \)

So then, the abstraction, a single vertical strand (not necessarily a straight line), with a pair of indicies as labels at either end is an element of \( I_n \), the indicies are from the set {1, 2, 3, . . . , n}.

Now we want to represent crossed (resp. uncrossed) pairs of strands, and it would be nice if we could also represent which strand crosses over which . . .

 

Suppose n = 2, then \( i,j \in {1,2} \). Then \( I_2 \) looks like:

\( \begin{pmatrix} \delta^1_1 &\delta^1_2 \\ \delta^2_1 & \delta^2_2 \end{pmatrix}\;=\; \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).

Now we write it like:

\( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}_{ij} \;=\; \delta^i_j \).

The abstraction as a strand with indicies (a diagram) is equivalent to this function that returns a 1 if two numbers are equal and 0 otherwise.

Kauffman: "In general, a tensor-like object has some upper and some lower indices. These become lines or strands emanating from the corresponding diagram".
From this we can take that the identity matrix with row and column indicies is a diagram with an upper and a lower strand, and that the Kroneckcr delta has no diagram, its upper and lower strand is the same strand.

We might consider the single strand with indicies (sorry about the Brit spelling), represents an abstract connection between the pair of index variables; if we think about the strand as representing an actual distance, then it can only represent that the pair of numbers is always distinct or separate, even when equal in value. And the rows and columns of a matrix are always distinct.

So a pair of strands has two pairs of index variables, shuffling these around is the method Kauffman uses to represent crossed and uncrossed pairs (of what seem to be abstract relations between pairs of indicies).

With a crossbar switch model of crossings, the variables label pairs of inputs and outputs, there is a notion of process or transmission of information of some kind from inputs to outputs; the labels are variables that "run" over some index set. We go to the Kronecker delta model when the inputs are say, upper indicies and the outputs are the lower indicies.

Say we have a,b as inputs and c,d as outputs, we then represent our "switch" as follows:

\( \delta^a_c \delta^b_d \) represents two (uncrossed) vertical strands, the switch is off.

\( \delta^a_d \delta^b_c \) is the same pair of strands but crossed over, the switch is on.​