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Friday, September 23, 2016

Very Cool.

Very Cool.

Originally shared by Richard Green

Knot mosaics

This picture by Samuel J. Lomonaco Jr and Louis H. Kauffman gives some examples of knot mosaics. A knot mosaic is the result of tiling a rectangular grid using the 11 types of symbols listed in the diagram, in such a way that (a) the connection points between adjacent tiles are compatible, and (b) there are no connection points on the outer boundary of the rectangle.

A natural combinatorial question is: how many ways are there to inscribe a knot mosaic in a rectangular grid of a given size? To make things slightly easier, we may assume that the grid is an n by n square, rather than an m by n rectangle. For a 1 by 1 grid, we have no choice other than to use the blank tile. For a 2 by 2 grid, we can either use four blank tiles, or we can arrange four tiles to draw a circle in the middle of the grid. For a 3 by 3 grid, there are a lot more possibilities: it turns out that there are 22 in all. As n increases, the number of knot mosaics grows very quickly. The first few values are 1, 2, 22, 2594, 4183954 and 101393411126.

To get an idea for how quickly this sequence might be growing, consider the much easier problem of filling a square grid with tiles that do not have to fit together compatibly at their edges. There are 11 types of tiles, and n^2 spaces in the grid, so the total number of ways to fill the grid is 11 to the power n^2. The great majority of these do not lead to knot mosaics, but the recent paper Quantum knot mosaics and the growth constant by Seungsang Oh ( proves that there exists a constant δ, called the knot mosaic constant, with the property that the number of knot mosaics in an n by n grid, for large n, is approximately δ to the power n^2. Because there are 11 types of tiles, it follows immediately that δ must be less than 11. The paper gives a much better estimate, namely that δ lies between 4 and 4.303.

Although knot mosaics might seem like an isolated curiosity, they have applications to knot theory and quantum physics. If one imagines the left and middle knot mosaics in the diagram as physical loops made out of an elastic material, it is possible to see that it would be possible to deform one of these links into the other without breaking the material. Mathematically, there is an operation on knot mosaics that has the effect of interchanging these two knots. However, the third knot turns out not to be equivalent to the others, in a way that can be made mathematically precise.

In their 2008 paper Quantum Knots and Mosaics ( Lomonaco and Kauffman use the knot mosaics as a basis for a Hilbert space called the quantum knot state space. The paper defines a group of symmetries generated by mosaic versions of the Reidemeister moves and mosaic versions of planar isotopies, and this group acts on the mosaics. The Hilbert space is meant to capture the quantum embodiment of a closed knotted physical piece of rope, and the group represents all the possible ways to move the rope around, without cutting it or letting it pass through itself. However, unlike classical knotted pieces of rope, quantum knots can model superpositions of several pieces of rope, as well as quantum entanglements.

Relevant links

The picture comes from the arXiv version of Lomonaco and Kauffman's paper.

The On-Line Encyclopedia of Integer Sequences has a longer list of values for the numbers of n by n knot mosaics. This one goes up to 11:

Here's another post by me with more details on the Reidemeister moves:

#mathematics #scienceeveryday

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