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Zombies and knots

Constructing the Jones polynomial to save the world

Day 0: smalltowns-ville, IA:

A man picks up his daily slice of breakfast pizza from his local gas station. What he doesn’t know is that it’s his last. By mid-day he feels terrible, by the time he’s ready to go home for dinner he’s already feasting on brains.

Day 4: CDC Headquarters

The Z-virus has spread midwest-wide. You’re working at the CDC as an expert in microscopy. You’re working frantically to get any information on the Z-virus you can. You decide to image the DNA of the Z-Virus.

DNA

Deoxyribonucleic acid (abbreviated DNA) is the molecule that carries genetic information for the development and functioning of an organism.

DNA is made of two linked strands that wind around each other to resemble a twisted ladder — a shape known as a double helix.

Macro Structure

Circular DNA

Dulbecco and Vogt (1963) and Weil and Vinograd (1963) discovered that double-stranded DNA of the polyoma virus exists in a closed circular form.

“Supercoiled” DNA

Vinograd, J., Lebowitz, J., Radloff, R., Watson, R., & Laipis, P. (1965) discover that double-stranded DNA can “supercoil”.

Day 7: CDC Headquarters

The spread is now nation wide but still under some control.

You’ve successfully imaged the DNA of the Z-virus and found DNA with a knot. Your CDC coworkers are using your findings to construct an anti-Z-virus. The anti-virus is the mirror of the DNA knot you’ve found. This will allow the human body to build anti-bodies for the Z-virus.

The CDC now needs you to verify that the DNA knot they’ve produced truly is the mirror of the Z-virus.

Anti-Knot

Mathematical Knots

“A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\R^3$ (or the 3-dimensional sphere $S^3$ ).”

Diagrams for knotted dna

$\to$

$\to$

Knot Equivalence

Reidemeister moves

Type I

Type II

Type III

Equal?

Playing with diagrams

What’s the important information inside a knot diagram?

Clockwise

anti-clockwise

Pictures are hard lets leverage algebra

Skein Relation

What are we looking for?

We want to use our bracket to build a polynomial that can tell two knots apart. In particular, we want to differentiate a knot and its “anti-knot”(mirror).

Putting pieces together

• How can we tell two knots apart?
• How can we use that and our bracket to build our polynomial?

Check what happens under Reidemeister moves

If our bracket “respects” Reidemeister moves it respects knot “equivalence”.

Type II

A problem

Back to computing

What we wanted

What we have

So we need

Putting pieces together

How can we select $A,\ B$, and $C$ to get equality?

Type II

Type I

Why is this a problem?

What do we have so far?

For Type II and III everything “works” with the rules:

Day 53

Time is running out. With your preliminary results in hand the vaccine is being produced. The future of the world is now on your shoulders waiting for your results.

How can we fix Type I

Orientation of a crossing

Writhe of a knot

The writhe $w\LP P\RP$ of a diagram $P$ of an oriented link is the sum of the signs of the crossings of $ P $.

Fixing Type I

What do we have?

For Type I, II, and III everything “works” for the polynomial $$V\LP P\RP=-A^{-3w\LP P\RP}\LA P\RA$$ with the rules:

We can now compute

Anti-Knot

Day 121

With the successful completion of your work the vaccine is being administer world wide. The President congratulates you for your work and the world is optimistic.

Day 300

The virus is completely controlled and you win every prize in every field imaginable!

The Jones Polynomial

The Jones Polynomial $V\LP \mathscr{K}\RP$ of an oriented knot $\mathscr{K}$ is the Laurent polynomial with integer coefficients in $t^{1/2}$.

Defined by $ V\LP \mathscr{K}\RP=\LP\LP-A\RP^{-3w(P)}\LA P\RA\RP _{t^{1/2}=A^{-2}} $ where $P$ is any oriented diagram for $\mathscr{K}$.

Worksheet