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.