MA 391 Composition and Communication

Knots and crossing numbers A knot is a simple, closed, non-self-intersecting curve in \(R^3\). It is natural to think of a knot as constructed from a string glued together at the ends, usually tangled in the middle. The knot for my project is knot 9-1 in the knot atlas, which is centrosymmetric. The crossing number of the knot is 9. It is common to use a knot diagram or 2D projection to graphically represent knots. A knot is represented by a curved line on the page. When the knot passes under itself, a gap is used at the crossing. The projection of the 9-1 knot is shown in the following picture: Sequence of crossing switches for unknotting There are three types of simple allowed ways to deform knot diagrams by changing the number of crossings as shown in the following picture. The unknotting number of a knot is the minimum number of times to switch the crossing points to untie it. The 9-1 knot has the unknotting number \(4\). As shown in the fol...

 A knot is a closed loop which is a transformation of the circle into 3D space! Unlike a physical knot, a mathematical knot has no ends, and thus isn't tied and can't be untied. The fundamental knot is the circle, which is the unknot.  For a knot more complex than the circle to be formed, the threads of the knot must be crossed a finite number of times. These can be undone through the Reidemeister moves, which move threads past each other. Certain operations do not change the not, and these create invariants of a knot.  It cannot be trivially proven that a given knot is not actually a clever twist of the circle. Coloring the knot could reveal if a not is not the unknot. However, the failure of coloring a certain number of times does not always mean a knot is the unknot, as different starting points for the coloring may affect the outcome as well as using more colors. The simplest coloring is three-coloring. For this process, three colors are used and the thread...

Introduction Somehow it's the final blog of the semester! This time we're talking about knots. Knots are a topic in mathematics, specifically topology. Knots are exactly what they sound like; you make a knot when you tie your shoes, for example. The only difference is that mathematical knots have the ends of the strings tied together and they cannot be undone. We have simple knot examples, like the trefoil knot. This is a knot of three crossings. We can get even crazier by looking at knots of 8+ crossings, which will be discussed in the rest of this post. (Image from centerofmathematics.blogspot.com) Crossing number Consider this knot. It has 10 crossings and is knot 10-139 in the knot atlas found at http://katlas.math.toronto.edu/wiki/10_139. Unknotting number The unknotting number is the sequence of crossing switches that give the unknot. From the knot atlas, I know the unknotting number of 10-139 is 3. This means that...

  There is even a mirror to help see the illusions!

  They have been removed to make space for the do over projects!

When you think of a knot, you may think of tying a string, rope, or shoe. But those knots are a little different than the mathematical knots we will be discussing today because those all have ends to the knot. A knot defined in mathematical terms is a closed curve in a 3D space that doesn’t intersect itself. The knot I will be focusing on in this blog post is knot 8_18 (according to Rolfsen Knot table from The Knot Atlas). The 3D print of the knot looks like this. We represent knots in a 2D space by drawing knot projections. One knot can have multiple projections, but here is the projection of the knot we will be working with. (http://katlas.math.toronto.edu/wiki/8_18) We can determine how many crossings a knot has by looking at its projection. A crossing is the place where a knot crosses itself in a knot projection. Therefore, this is an eight-crossing knot, hence why it is labeled knot 8_18.  Now that we know the basic information for this knot, we will focus on determining the ...