Thompson's Group F and Knot Theory

This repository contains scripts related to links generated by elements of Thompson's group F.

Background

For the purposes of this repository, an element of F can be thought of as a pair of binary trees (rooted, with each node having zero or two children) with the same number of leaves. These are often drawn with the second tree inverted so that its leaves lie on the same points as the first. For example, the following represents an element of F with 5 leaves:

Since this diagram is rather unwieldy, it will be simpler to represent the above diagram as the following "fraction" of bracketings: ((o(oo))(oo)) / ((o(o(oo)))o) The left-hand string represents the top tree; the right-hand, the bottom. Here, the os represent leaves of the trees, and a pair of parentheses represents a subtree. To be annoyingly precise, here's a context-free grammar for the setup:

    TREEPAIR ::= TREE "/" TREE
        TREE ::= "o" | "(" TREE TREE ")"

We often only consider those tree-pairs which are "reduced", meaning the top tree cannot have a caret (represented by the exact string (oo)) spanning the same two leaves as a caret of the bottom tree. See OEIS. Each element of Thompson's group F has a unique representation as a reduced tree-pair.

In his 2018 paper (and prior in a 2014 paper), the late Vaughan Jones describes a function from the set of tree-pairs to the set of links: edges of the diagram become strands of the link, and new strands are added to each face, connecting a caret of the top tree to a caret of the bottom. Crossings are formed by an over-strand connecting each internal node's two children and an under-crossing connecting the node's parent to the newly added face strand. Below is the output of this mapping on the tree-pair from before (the result is a two-component unlink):

Jones showed that all links arise from elements of F, in analogy with Alexander's similar theorem about the braid group. It would be desirable to classify which elements of F give the same links, perhaps finding a sufficient set of "moves" in analogy with Markov's theorem aboud braids.

The code here

These are the most important files here:

thompson.py

This file considers strings like "((o(oo))(oo)) / ((o(o(oo)))o)", constructs a data structure that represents the resulting link, then computes the signed Gauss code of the link (a format more easily understandable by SageMath).

link_classification.py

This file uses thompson.py and interfaces with SageMath to classify all reduced tree-pairs by the HOMFLYPT polynomial of the link they produce. To use it, type somthing like this into a working SageMath REPL:

import sys
sys.path.insert(0, "[PATH_OF_THIS_REPO_GOES_HERE]")
from link_classification import main, words_to_link
from time import time

t0 = time()
mapping = main(6)
t1 = time()

with open("unoriented6.txt", "w") as f:    
    for key, arr in mapping.items():
        L = words_to_link(*arr[0])
        az = L.homfly_polynomial('a', 'z', 'az')
        print("Description:", file=f)
        print(f"Components: {key[0]}", file=f)
        print(f"Homfly (lm): {key[1]}", file=f)
        print(f"Homfly (az): {az}", file=f)
        for pair in arr:
            print('\t' + ' / '.join(pair), file=f)
        print(file=f)
    print(f"elapsed time: {t1 - t0}", file=f)

The output of this is a file like unoriented_data/unoriented5.txt. Note that unoriented9.txt took 5 hours to compute on my laptop, with most of that time being dominated by the computations of HOMFLYPT polynomials.

moves.py and moves_to_components.py

This is an attempt at implementing several known link-preserving "moves" on tree-pairs to see how far they get towards being "all of the moves we need". Tests are included in test_moves.py and can be run with PyTest.

Results

Rushil, Raghavan and Dennis Sweeney. "Regular Isotopy Classes of Link Diagrams From Thompson's Groups." ArXiv:2008.11052 [Math], Nov. 2020. arXiv.org, http://arxiv.org/abs/2008.11052.