A stochastic interpretation of knot invariants

What do knot invariants of the type introduced in invariant have to do with stochastic processes? As I shall discuss in this article, it turns out that a stochastic viewpoint of invariants is incredibly useful for understanding their properties, as well as deriving more general invariants.

The idea of considering random walks on knots, braids and tangles was proposed in [1]. Given a knot, the authors deduced properties about this knot by imagining a particle traversing on the strands. Key in this interpretation was the possibility for the particle jump from strand A to strand B when strand A crosses B. This process can be described by considering the transition matrix of a particular Markov chain.

In this post, I’ll give a description of the main ideas in [1]. I’ll assume that you know the definition of a Markov Chain, the Burau representation, as well as the information from the previous sections.

Jones’ bowling alley#

Before “jumping” into the nitty-gritty, let us start with some formalities. Recall that the Burau representation associates to every braid on $n$ strand, a $n \times n$ matrix with entries that a rational function in a variable $t$ . In particular, the Burau representation sends every braid group generator $\sigma_i$ (the crossing of strand $i$ over strand $i+1$ ) to a matrix that acts as the identity everywhere except in $2 \times 2$ block:

The stochastic ideas in [1] are fundamentally based on Jones’ bowling alley originally introduced in [2], which the paper quotes:

“For positive braids there is also a mechanical interpretation of the Burau matrix: Lay the braid out flat and make it into a bowling alley with $n$ lanes, the lanes going over each other according to the braid. If a ball traveling along a lane has probability $1 − t$ of falling off the top lane (and continuing in the lane below) at every crossing then the $(i, j)$ entry of the (non-reduced) Burau matrix is the probability that a ball bowled in the $i$ th lane will end up in the $j$ th”

In this view, the Burau matrix (or representation, if you must) is rather a table that records probabilities. Indeed, upon careful inspection of the Burau matrix, we notice that

1. $t \in (0,1)$ , the numbers $t$ and $1-t$ are probabilities;
2. every entry is nonnegative;
3. the rows sum to 1.

These are precisely the properties that a stochastic matrix should have. Thus, what we obtain is a Markov chain! In probabilistic language, Jones interpretation reduces to the observation that the Burau matrix is a stochastic matrix, and a bowling ball is running one step of a Markov chain on the lanes.

Let’s consider this more graphically. Figure 1 shows a single positive crossing drawn as a bowling-alley junction.

From braids to string links#

Considering braids is rather restrictive, as every strand always heads upward. In [1], this restriction is relaxed by considering so-called string links. These are defined as follows.

An example of a string link has been shown in Figure 2.

A random walk on a string link#

Instead of thinking about Jones’ bowling alley, we imagine a walker moving up from a point $i \times 0$ along strand of a given string link $\sigma$ to a point $j \times 1$ . In doing so, the following rules need to be satisfied:

1. The walking direction should always be in agreement with the orientation of strands of $\sigma$ .
2. If the walkers arrives at a crossing on the lower segment, they keep walking on the lower segment passing through that crossing.
3. If the walker arrives at a crossing on the upper segment, they may choose either to jump down walking on the lower segment or keep walking on the upper segment passing through that crossing.

Any such route from $i\times 0$ to $j\times 1$ is called a path. A piece of a path that returns to where it started is a called a loop. A path is called simple if it contains no loops. There are only finitely many simple paths.

Given a path, the walker made decisions about whether they jumped down or kept on walking. To track which decision have been made, we associate to every path $P$ a weight $w(P)$ . These weights depends on the sign of the (potentially many) crossing that are encountered when walking over a chosen path. Hence, we need to pin down what positive and negative mean. Orient both strands upward and look at which way the over-strand (the one drawn unbroken) runs: if it goes from bottom-left to top-right, the crossing is positive. On the other hand, if it goes from bottom-right to top-left, it is negative. This has been illustrated in Figure 3.

Now, the weight of a path is defined as the product of weights collected at every decision point (writing $\bar t = t^{-1}$ ):

• at a positive over-crossing: weight $1-t$ if the walker jumps down, weight $t$ if they stay up;
• at a negative over-crossing: weight $1-t^{-1}$ if the walker jumps down, weight $t^{-1}$ if they stay up.

This results in a matrix $B(\sigma)$ assigned to the string link $\sigma$ , whose $(i,j)$ entry is the sum of the weights of all paths from $i\times 0$ to $j\times 1$ :

and it’s $0$ if no path exists. This matrix is called the Burau matrix of the string link $\sigma$ . For a braid with only positive crossings, the choices “stay with probability $t$ , fall with probability $1-t$ ” correspond precisely to Jones’s bowling alley notion.

An example#

Consider the simple two-strand string link $\sigma$ shown in Figure 1. By considering simple paths from sources to sinks, we find weights

• source 1 to sink 1: one simple path, $w=1$ ;
• source 1 to sink 2: one simple path, $w = 1-t$ ;
• source 2 to sink 1: one simple path, $w = \bar t(1-t)$ ;
• source 2 to sink 2: two simple paths, $w = t$ and $w = (1-t)^2(1-\bar t)$ .

Every path can be combined with any number of copies of the loop, and the loop sum is geometric:

Then, according to the definition above, we find

Notice that $(1,1,\dots,1)^T$ is fixed — the rows are arranged so that $1$ is always an eigenvalue.

Markov Chains#

Now, going back to the stochastic view, two properties of $B(\sigma)$ are to be expected:

1. If $\sigma$ has a single strand, all paths land at the same sink, so $B(\sigma) = 1$
2. The rows of $B(\sigma)$ sum to 1.

These two properties are rigorously proven in [1]

For $B(\sigma)$ to truly be a stochastic matrix, one requires it to exclusively contain nonnegative entries. This can only be the case for positive string links, which are string links that are exclusively built up from positive crossings. In that case, every weight results from genuine probabilities $t$ and $1-t$ . In the theory of Markov chains, any stochastic matrix is the transition matrix of some Markov chain. In our case, this chain can be made explicit:

• the state space $S$ is the set of components (strands) $\{1,\dots,n\}$ of the string link;
• the transition probabilities are $p_{ij} = B(\sigma)_{ij}$ , the total weight of all paths from source $i$ to sink $j$ ;
• a sequence of states $X_0, X_1, X_2, \dots$ records which component the walker walks on at each time step, with the defining Markov property that the next state $X_{m+1}$ depends only on the present state $X_m$ , not the past.

An example of a Markov chain#

For a positive string link the picture becomes fully stochastic. The string link in Figure 2 is not positive as the left-most crossing is a negative one. Let’s replace this crossing with a positive one. We obtain a two-strand link whose Burau matrix is equal to

Every entry is nonnegative and each row sums to $1$ , hence this is a true stochastic matrix. This gives rise to a Markov chain

Now notice that this Markov chain describes a single traversal, not an iterated process. For this to occur, a walker that ends up at the top of the string link has to be able to move to the beginning again, i.e. if sink $j$ at the top is identified with source $j$ . This identification can be made upon closing the string link: we glue the top to bottom, $j\times 1 \leftrightarrow j\times 0$ . This turns the strip into an annulus.

Closing words#

The stochastic viewpoint is rather crucial, as it has recently been extended in the context of perturbed Gaussians by Bar-Natan and Van der Veen in [3]. This view has resulted in efficient methods to compute strong knot invariant. I will go into the details of the stochastic view of this model in a future post.

References#

1. X. Lin, F. Tian, and Z. Wang, Burau representation and random walks on string links, Pacific Journal of Mathematics, vol. 182, no. 2, p. 289–302, 1998. doi:10.2140/pjm.1998.182.289
2. V. Jones, Hecke algebra representations of braid groups and link polynomials, in New Developments in the Theory of Knots, World Scientific, 1990, vol. Volume 11, p. 20–73.doi:10.1142/9789812798329_0003
3. D. Bar-Natan and R. van der Veen, A perturbed-Alexander invariant, Quantum Topology, vol. 15, no. 3, p. 449–472, 2024. doi:10.4171/qt/206