Knot Theory

by Erin Gottschalk, Simon Johnson, Meghan, Elizabeth Nguyen, and Brooke Rogers*.

*Brooke helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

What We Did:

Knot theory has many different applications in math including algebra and geometry, and (outside of math) physics. We learned that we can use algebraic techniques to describe knots. When trying to understand knot theory we learned that it is very helpful to work in a group and read the definitions out loud. Us working together was key in understanding knot theory.

What Are Knots & Links:

A knot is a loop in a three dimensional space that does not intersect with itself. You could tie a piece of string and make it into a knot, right now. A link on the other hand is several knots that intersect each other. Knots can be very deceiving, they may not look like it but they could be the same knot. There are a few different types of knots that are the same but don’t really look like it.  A knot is equivalent to another knot when one can e manipulated to become the other knot. We learned a few simple ones…

Unknot: this knot can be shown in many different ways.

165px-Unknots.svg

Left-handed and right-handed Trefoils:

108-1089032_the-two-trefoil-knots-left-handed-trefoil-knot.png

Hopf Link:

1Hopf link

 

What Is A Knot Diagram:

These knots (and links) are being drawn as a knot diagram. A knot diagram is essentially a drawing of a knot where the connections of the strings are separated, it’s like if you were to shine a light above the knot and trace the shadow.­­

Reidmeister Moves: There are three Reidmeister moves, they are used to untie knots. A knot is equivalent to another knot when one can e manipulated to become the other knot by using only the three Reidmeister moves.

ReidemeisterMoves_1001

 

The images featured in this post are from:
http://mathworld.wolfram.com/AmphichiralKnot.html
https://en.wikipedia.org/wiki/Unknotting_problem
https://commons.wikimedia.org/wiki/File:The_two_trefoil_knots.pdf
http://mathworld.wolfram.com/HopfLink.html
http://mathworld.wolfram.com/ReidemeisterMoves.html

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