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Abstract: Knots appear everywhere in life - most familiarly, on shoelaces and ropes and jewelry, but they also hide in our DNA and lie behind some very interesting science in areas like polymer physics and quantum cryptography. We can study the knot as a mathematical object and ask all sorts of interesting questions about it. Sometimes we want to classify knots: If I draw a knot, and you draw a knot, when should we say they are the same? If they are different, how do we prove that? Can I list all the knots, at least all the ones that are "kind of small" (whatever that means)? Often we define and study things called "invariants", numbers or polynomials or other things that we can calculate for a knot and which help us predict their properties or distinguish them from one another. Occasionally we are more interested in the effects of the knots on whatever is being knotted: If your DNA is twisted up like the knot I just drew, will that affect your gene expression? Or, if I pick the wrong knot when I engage my quantum cryptographic security, will you be able to read my email? Finally, sometimes, we just want to know how common a knot is, or how likely it is to have certain properties, which can have far-reaching implications for how useful some of these properties are or how the knotted things in the world behave. We will take a brief trip through knot theory and talk about all of these questions, culminating in a quick look at a basic example: if I randomly draw a knot, how many separate pieces of string will it have? (The experts might rightly object to this extremely casual terminology; a more precise way to phrase it could be: We will derive a generating function describing the distribution of the number of components of a randomly generated link grid diagram.)
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