Easy cases of the volume conjecture?

 November 18, 2013 mathematics

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The volume conjecture relates the hyperbolic volume of a knot complement to quantum invariants of the knot. Specifically, the conjecture is that

\operatorname{Vol}(S^3 \setminus K) = \lim_{N \to \infty} \frac{2 \pi \log |J_N(K;\xi_N)|}{N}

where J_N computes the colored Jones polynomial and \xi_N = e^{2\pi i / N} . For some knots K , there are nice formulas for J_N(K;\xi_N) . For instance, if K is the figure eight knot, then J_N(K;q) can be written as

J_N(K;q) = \sum_{j=0}^{N-1} \prod_{\substack{k=-j \\ k \neq 0}}^j \left(q^{(N-k)/2} - q^{-(N-k)/2}\right).

When q = \xi_N and one takes the limit, this sum transforms quite nicely into the Riemann sum which computes 6 \,\Lambda(\pi/3) , which is the hyperbolic volume of the figure eight knot complement.

Are there other cases in which one can verify the volume conjecture directly by finding a nice form for the colored Jones polynomial? Yamazaki-Yokota in “On the limit of the colored Jones polynomial of a non-simple link” verified, in the same direct way, the volume conjecture for a certain link with volume 6\,\Lambda(\pi/3)+16\,\Lambda(\pi/4) . I hope one could find other examples by searching for links and knots with “nice” volumes in terms of the Lobachevsky function \,\Lambda .

From my search with SnapPy,

Of course, there are lots of nonhyperbolic knots and it is hard to prove the volume conjecture even for those, in spite of the fact that their hyperbolic volume is “really nice” (namely, zero!).


Reflecting Triangles, live

 February 23, 2011 personal mathematics

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A while back I made some movies which began with a triangle in the plane, reflected that triangle through its three sides, reflected those triangles through their sides, and so forth. The interesting result is that for only four shapes of triangles, the resulting set of triangle vertices is discrete.

Using Raphael and a plane geometry package that I wrote, I quickly redid this visualization in Javascript; you can now move the vertices around to see the effect on the reflected triangles.


Culturomics

 December 18, 2010

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I have really fallen in love with Google Books Ngram Viewer, so I thought I’d do a little ``culturomics" myself. Here’s an image I made using Google’s data:

Numbers in Print

The brightness of the pixel at position (x,y) is related to how frequently “ x ” appears in books published in the year y . Specifically, if p is the number of times “ x ” appears in print during year y , divided by the number of times any number less than 2100 appears in print during that year, then (1 - p)^{1500} is the brightness of the pixel at (x,y) .

The dark, diagonal edge along the right hand side appears because in year y there are many published appearances of numbers near y .

Dark diagonal edge

World events have left their mark on the numbers appearing in books! For example, 1914 is still being talked about long after 1914, as evidenced by the darker line above 1914.

If we look at numbers just above 1000 and turn up the contrast a bit,

Around one thousand

we see an echo of the dark diagonal, from people writing (or more likely, the OCR software reading) zero instead of nine in the year. There’s a dark column for the Norman conquest in 1066; a number like 2^{10} = 1024 was not so important until the 20th century.

If we look at numbers just above 1300,

Above 1300

we can see an diagonal line from 1800s being read as 1300s, and a dark vertical line above 1453 (the “end” of the middle ages). In the 18th century,

Above 1700

1776 is quite visible. And finally, a puzzle:

Why 2044

Why was “2044” so significant until the 1920s?

2043,2044,2045 in Google ngrams viewer

I’d love to know the answer to this question. The only thing I can guess that might relate the year 1919 to the year 2044 is solar eclipses.


Many more Lights Out

 July 17, 2010 mathematics

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A very long while ago I posted some solutions to Lights Out; back then, I solved the n -by- n board by row-reducing an n^2 -by- n^2 matrix.

Since then, both Boris Okun and Brent Werness pointed out to me that I should’ve solved Lights Out by using a scanning algorithm: propagating the button presses down one row at a time, and exponentiating the propagation matrix to make sure that I don’t get stuck at the last row.

This is much faster.

With this method, here is a (scaled down, auto-leveled) 2000-by-2000 solution:

Solution to 2000x2000 Lights Out

And here is a (very much scaled-down, auto-leveled) 5000-by-5000 solution:

Solution to 5000x5000 Lights Out

Reflecting Triangles

 March 16, 2010 personal mathematics

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My advisor, Shmuel Weinberger, was teaching Math 113, and asked for some pictures of the following procedure:

  • Start with a triangle in the plane.
  • Reflect that triangle across its three sides.
  • And repeat, reflecting the resulting triangles through their sides, and so forth.

I made a couple movies of this, illustrating this procedure as you move through the space of triangles. Observe how, for only four shapes of triangles, the resulting set of triangle vertices is discrete.

Movie with only a few triangles


Projector on Blackboard

 January 19, 2010 mathematics

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I recently gave a beamer talk, which gave me the chance to point the beamer at my blackboard.


My mathematical genealogy

 June 11, 2009 general

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According to the Mathematics Genealogy Project, my mathematical genealogy is:

There are some branches to choose among, but I think the branch starting with Pacioli is the most appropriate.


Möbius strip, and pairs of points on a circle.

 January 28, 2009 personal mathematics

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Here’s a little movie I made:


I can drive!

 September 26, 2008 personal

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I took my road test this morning—and I passed!

After all these years, I am a licensed driver. Now, where should I drive to?


Global Warming according to Google

 August 22, 2008 personal

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Google Trends plots the search volume (or some other measure? search percentage?) for a given phrase over time. It’s ridiculously fun!

As an example, let’s look at the number of times people search for the words hot and cold. I downloaded the CSV file offered by Google trends to make the following graph:

The thick red and blue lines are the linear regressions on the number of searches for hot and cold, respectively. Behold!—people are searching more often for hot lately, and less often as of late for cold! The search volume does seem to be related to the temperature: you might notice that the search volume for cold dips under the regression line during the summer, but exceeds it during the winter.

And so, global warming is being revealed in our search habits. Maybe I should’ve titled this post “Google warming.”