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.