Linking back to Links

Continuing on, we talk about the invariant made by links, and the analogous unlink (to the unknot).

Because mathematicians are always very creative people, things get names like as "the unknot" and "the unlink". If you remember the unknot (hint, it's the individual knots on the left and right in Figure 1), the unlink simply consists of two unknots (see Figure 1).

Alternatively, the Hopf link also consists of two unknots, but joined together (see Figure 2).

Figure 1 - The (splittable) unlink is made up of two unknots.

Figure 2 - The (unsplittable) Hopf link is also made up of two unknots.

At this point, you may think, well, it would be nice if we had a way to show exactly how linked two things were. Well, surprise, there is -- and it's called the linking number.

The way to find the linking number is, if you take a 2 component link and designate one as being on the left, and the other being on the right. Then, look at all the places the two cross. If the left one crosses over the right, add one to your total, and if the right one crosses over the left, subtract one from your total (Figure 3).

(+1)

Figure 3 - Calculating Linking Number

Then take this and divide by two. You have your linking number.

If you switch which component is on the left and right, you'll get either the positive or negative of the value. This is an invariant, because, no matter how you twist or turn the knot, the linking number will stay the same.

Unsurprisingly, the linking number of the unlink and the unknot is zero. Also, it is unaffected when the Reidemeister moves (from last week's post) are implemented onto the link.

Anyways, that's it for now. Thanks for keeping up, and see you next week!



References:

Adams, Colin Conrad. The Knot Book. New York: W.H. Freeman, 1994. Print.

Bio.math.berkeley.edu,. N.p., 2016. Web. 27 Feb. 2016.

Exploring Optimal Nutrition,. "Exploring Optimal Nutrition". N.p., 2016. Web. 27 Feb. 2016.

Ndstudies.gov,. "3 - Venn Diagram | North Dakota Studies". N.p., 2016. Web. 27 Feb. 2016.

Upload.wikimedia.org,. N.p., 2016. Web. 27 Feb. 2016.

An Intermission, Papers, and Magnetic Fields

This is a brief intermission between the first and second "explanatory" posts of today.

My posts so far have all been about knot theory basics (getting into it, etc), so I realized it would probably be good to talk a bit about what I've been doing.

So far, in the past couple weeks, I've been finishing up on going through various knot invariants, and their properties, some of which I've posted about, and some of which I have yet to post about, in order to familiarize myself with their properties.

I will also be starting to look at a few papers on various invariants mentioned in the book I've been reading.

In additional to my project, by suggestion of my SRP mentor professor, I go to the weekly geometry seminars. They always have interesting topics, that are difficult but interesting to try to understand. Today's was on Microscopic Electromagnetism and Differential Forms. It was a derivation, using vector fields and wedge product to derive Maxwell's Equations (electric/magnetic fields, potentials, sources - all that fun stuff).

My posts will probably be mostly along the lines of explaining knot theory, with once in a while check-ins with the status of my project.

Hope you guys keep reading, and keep up with my posts!

Linking Venn Diagrams to Paper Clip Chains

Hello! Thanks for reading and keeping up with the posts here.


So far, in the course of these posts, we've only been talking about or single knotted loops of string.

Today, we'll be talking about links, or multiple knots mixed together.

If you have ever made paper clips chains or have drawn a Venn Diagram, think of it as something like that, except with the loops of knots (Figure 1, 2).

Figure 1 - A triple diagram. (Or, as a link, famously called the Borromean Rings.)

Figure 2 - A paper clip chain - Links are like this in that they are looped in eachother.

The diagram in Figure is be an example of the Borromean Rings link. Another example of a link, would be the Whitehead Link below

Figure 3 - The Whitehead Link is made up of two separate loops.

We say a link has n components if it is made up of n different parts. For example, the Borromean Rings link in Figure 1 would be composed of three components.



Links can be classified as splittable and non-splittable. Splittable links are those you can twist so that the pieces become separate.

Think of this as taking two rubber bands and trying to roll them together (we've all done it...). No matter how you try to push them together, you can still pull them apart (move them around) so that they're back to two separate rubber bands.

Splittable links are kind of like this, except, instead of just rubber bands, you can take knots and try to smush them together. The figure below shows the two examples.

Figure 4  - Two different splittable links. 

We mostly focus on unsplittable links (such as the Whitehead Link in Figure 4), as splittable links can be pulled apart and considered as two separate knots.

That's it for now (as this has been a pretty long post), and we'll continue after a short intermission!

Reidemeister Moves

Hello!

For the second part of our two-post series, we'll be talking about very special kinds of "moves" in knot theory, with which you can pull and twist a knot around while keeping it as the same knot. These rearrangements, as a whole, which are made without passing the knot through itself, are called isotopies.

There are three types of moves allowed, called the Reidemeister Moves (see Figure 1). These moves enable us to take away or add a crossing.

Figure 1 - The Reidemeister Moves

The first Reidemeister move allows us to put or remove a twist in the knot (Figure 2),

Figure 2 - This is the first Reidemeister move. 

The second Reidemeister move allows us to either add two crossings or remove two crossings from the knot (Figure 3).

Figure 3 - This is the second Reidemeister move. 

The third Reidemeister move allows us to slide a strand of a crossing from one side to another of a crossing (see Figure 4).

Figure 4 - This is the third Reidemeister move. 

Invariants, which we talked about earlier, are preserved under Reidemeister moves, which is what makes these especially important.

Thanks for reading, and keep up for the next post!

Works Cited:

Adams, Colin Conrad. The Knot Book. New York: W.H. Freeman, 1994. Print.

Armstrong, John. "Reidemeister Moves". The Unapologetic Mathematician. N.p., 2007. Web. 20 Feb. 2016.

Win.tue.nl,. N.p., 2016. Web. 20 Feb. 2016.

Knot Composition and Knot Inverses

Hello!

Thanks for reading and keeping up with the weekly updates. The structure of these posts, for now, will be mostly review of different invariants, basics of knot theory, etc, before we get into more interesting stuff.


For the first part of our 2-post daily series, we'll be talking about composition of knots, or how you take two knots, and stick them together into another knot.

If you have two knots, M, and N, to compose them, you cut off a small arc of each knot, and join the two knots at the newly made four endpoints. This composition is denoted by M#N. (See Figure 1) 

Figure 1 - This is a composition of knots K1, K2, both the trefoil.

So now you may be wondering: Okay, great. You can take two knots (which you don't know a lot about) and squish them together into another knot (that you know even less about). What's good about that? 

Essentially, with these definitions, we can call a knot either "prime" or "composite." The knot is composite if it can be made as a composition of two nontrivial knots (fancy word for not the unknot), and it's prime if it cannot (see Figure 3). 

For example, the square knot below would be composite, as it is made up of the composition of a trefoil and its inverse. (More about that later) (see Figure 1,2)

Figure 2 - The square knot is composite.

The trefoil that it's made up of (pictured above in Figure 1) would be a prime knot. 

The inverse of a knot, as we talked about, earlier works like this. Consider the trefoil from earlier. An oriented trefoil (or an oriented knot) is a knot on which you choose a direction to move in (see Figure 3)

Figure 3 - An oriented trefoil knot.

The inverse of a knot is its mirror image.

A knot is called chiral if its inverse is not equivalent to itself. Alternatively, a knot is called amphichiral if it is equivalent to its to its inverse.

From here, then, we can go into invertible (reversible) knots, which means that the knot can be twisted around so that it appears the same, but with its orientation (see earlier) going in the opposite direction.

Thanks for reading this, now onto the next post about the Reidemeister Moves!

Works Cited

Adams, Colin Conrad. The Knot Book. New York: W.H. Freeman, 1994. Print.

Mathworld.wolfram.com,. "Knot Sum -- From Wolfram Mathworld". N.p., 2016. Web. 20 Feb. 2016.

Introduction to Invariants

Hello!

First of all, thanks for keeping up with this blog, and the uneven distribution of posts.

This post will mostly be explaining some of the basics of knot theory.

So starting off, we talked about how there are many, many different kinds of knots, in last weeks post, and how, they're all basically "a closed, non-self-intersecting curve that is embedded in three dimensions" (read: circles). 

From this, we can consider the projections of a knot. Projections are ways which we can display the knot, and each knot can have many, many different projections. Think of it as a knot in a bubble that we can turn to look at. For example, you can find different projections of the KotW (Knot of the Week) here: 8_17 Projections. (If you click around, you'll see that the knot looks different, but stays the same,)

Another interesting knot property to consider is the crossing number of the knot. A crossing is a place where a knot crosses itself, and the crossing number, naturally, would be the (minimum) number of times a knot would have to cross with itself.

Figure 1 - The crossings of this 7_5 knot has been circled. 

The knot has 7 crossings which cannot be simplified. So even if you take a part of the knot, and you keep on twisting and twisting and twisting it, the crossing number would not change.

The crossing number, here, is what we would call an invariant of a knot.

A knot invariant is a property of a knots that will stay the same even when it is pulled around. Because of this, knot invariants are good ways, and the typically used ways, to classify knots. There are many different invariants used, including knot polynomials, bridge number, curvature, etc.

Hope you enjoyed this tidbit, and expect another post very soon!



Works Cited:

Knotilus.math.uwo.ca,. "Knotilus". N.p., 2016. Web. 13 Feb. 2016.

Mathworld.wolfram.com,. "Knot -- From Wolfram Mathworld". N.p., 2016. Web. 13 Feb. 2016.

Knot of the Week

The Knot of the Week: The 8_17 knot.

This knot is the simplest of the non-invertible knots, which means that you cannot deform the knot (move parts of it around) so that you can make it the mirror version of itself.

If you want to learn more about it, feel free to go here: 8_17.

References:

Commons.wikimedia.org,. "File:8 17 Knot.Svg - Wikimedia Commons". N.p., 2016. Web. 13 Feb. 2016.