Monday, July 27, 2015

To Knot or Not to Knot

Today in Knot Theory we started learning about graph theory, and how it relates to knots. We started with some fairly simple problems, like the Konigsberg bridge problem, and proved (one of) Euler's (many) theorem(s), that graphs were Eulerian (meaning a path can be made using every edge exactly once and ending at the same point you started at) if and only if there are an even number of edges connecting to each vertex. We spent most of the rest of the time learning how to convert link diagrams into graphs and vice versa, because it's kind of complicated and gets messy very quickly when you look at larger graphs and knots.

The Konigsberg bridge problem
After lunch we learned about cheirality (also spelled chirality), and cheiral knots. The word comes from the Greek word for hand, cheir, and is based on the fact that a left hand and a right hand, though mirror images of each other, are distinct. If a knot is cheiral, then it cannot be deformed into its mirror image. This has applications in chemistry, because some molecules are cheiral. For example, Thalidomide was a drug developed around 1960 in Europe, and widely prescribed to pregnant women as a cure for morning sickness. The left-hand molecule of Thalidomide completely cured nausea, but its mirror image, the right-hand molecule, had very different properties and caused the side effect of severe birth defects. The drug was banned. Another example is limonene--one enantiomer (the term for the versions of a cheiral molecule) smells like oranges, while the other smells like harsh, fake lemon, or turpentine.

We also saw horses on our walk
In the TA session we played several games with knots. One of them was called To Knot or Not to Knot. It was a two-player game, and started with a link projection (a link diagram, but with none of the overcrossings or undercrossings shown). Each player on their turn made a crossing an overcrossing or undercrossing, but one was trying to make an unknot and the other was trying to make the diagram into a knot. (As it happens, the player trying to make an unknot always wins unless the player trying to make a knot takes their turn second on a link projection with an even number of crossings.) 

I'm not sure why there was a pancake...
This evening, Jenny took the four of us out for dinner to Puckett's BBQ. We stopped at a few stores as we walked towards the restaurant, and made note of an ice cream place we planned to get dessert at after dinner. At Puckett's, I had lemonade to drink, and they gave free refills. Somehow, I drank five glasses of lemonade. I'm sure at least one of those glasses was Hummd, and I will continue to swear that the reason the rest of those vanished so fast was that they were mostly ice anyway. I ordered the Gouda mac and cheese for dinner (so did Hummd and Arnold--it was irresistible) and it was absolutely amazing. First of all, mac and cheese is delicious anyway as long as the cook doesn't completely screw it up. With the rich taste and texture of the melted Gouda, it was mouthwatering. All baked in a cast-iron skillet on top of tender brisket with just a touch of barbecue sauce mixed in to bring out the smoky Gouda flavor, and it was positively heavenly. Yum!

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