Crossing matrices of positive braids

We think of a braid with $N$ strands as the homotopy class, modulo endpoints, of a geometric braid: an ensemble of $N$ differentiable paths $p_{i}\!\left(t\right)$, $i=1,\dots,N$ in the plane (the strands of the braid) whose tangent is never horizontal; the set of (distinct) initial positions $\{p_{i}\!\left(0\right)\ :\ i=1,\dots,N\}$ and the set of final positions $\{p_{i}\!\left(1\right)\ :\ i=1,\dots,N\}$ differ only in their common vertical coordinate, which we take to be $1$ for the initial and $0$ for the final points; it is not required that the initial and final positions of any particular strand be horizontally aligned. We assume for convenience that these paths are pairwise transverse, so that there are only finitely many crossings between strands, and each such crossing is assigned crossing data: it is regarded as positive (resp. negative) depending on whether it is left-over-right or right-over-left as we move down the path.

Given a geometric braid $b$, we define its crossing matrix as the $N\times N$ matrix $\boldsymbol{\mathcal{C}(b)}$ whose $i,j$ entry is the algebraic crossing number (number of positive crossings minus number of negative crossings) for strand $i$ crossing over strand $j$. By definition, the diagonal entries of $\mathcal{C}(b)$ are zero.

It is easy to check that the braid relations (which do modify some crossings) don’t affect the crossing matrix. So the crossing matrix $\mathcal{C}(b)$ can be thought of as defined on the braid represented by $b$. The description of a geometric braid in terms of crossings of strands is a point of view used by Thurston [4], and contrasts with the point of view of Artin (in terms of generators and relations) in his original expositions of the braid group $\mathfrak{B}_{N}$ [1, 2].

Attached to each geometric braid is the permutation on the set of horizontal coordinates of the starting points of strands which takes (the horizontal coordinate of) the starting point of each strand to its ending point.

We use Greek letters to denote permutations, regarded as rearrangements–that is, permutations act on positions rather than elements: a permutation $\pi\in\Sigma_{N}$ will be specified by the word²²2 Permutations will act on words on the right, denoted by superscript. $(12\cdots N)^{\pi}=\pi_{1}\pi_{2}\cdots\pi_{N}$ in the numbers $1,2,\dots,N$ resulting from the action of $\pi$ on the word $12\cdots N$. This contrasts with regarding a permutation as a bijective mapping $\pi\!\colon\!\left\{1,2,\dots,N\right\}\negthinspace\rightarrow\negthinspace\left\{1,2,\dots,N\right\}$ on a set of $N$ elements and denoting it by the $N$-tuple $(\pi\!\left(1\right),\pi\!\left(2\right),\dots,\pi\!\left(N\right))$ of images of the individual elements $1,2,\dots,N$ under this mapping. In fact the word $\pi\!\left(1\right)\pi\!\left(2\right)\cdots\pi\!\left(N\right)$ in our “rearrangement” notation denotes the inverse permutation³³3 The inverse of a permutation or braid will be denoted by an overbar: $\bar{\pi}$. . For example, the rearrangement $\pi\in\Sigma_{4}$ which acts on the positions $1,2,3,4$ via

takes the word $abcd$ to

while

We extend the “rearrangement” action of permutations to matrices: If $A=((a_{ij}))$ is an $N\times N$ matrix and $\pi\in\Sigma_{N}$, then $A^{\pi}$ is the matrix obtained by rearranging the rows as well as the columns of $A$ according to $\pi$.

When a matrix is the crossing matrix of a braid, say $A=\mathcal{C}(a)$, then the permutation $\pi_{a}$ associated to $a$ can be read off of $A$: every (positive) crossing of strand $i$ over another strand moves its position one place to the right, while every (positive) crossing of another strand over the $i^{th}$ moves it to the left one space. From this it follows that $\pi_{a}$ is defined (as a mapping $\pi_{a}\!\colon\!\left\{1,2,\dots,N\right\}\negthinspace\rightarrow\negthinspace\left\{1,2,\dots,N\right\}$) by adding to $i$ the sum of the $i^{th}$ row and subtracting the sum of the $i^{th}$ column of $A$:⁴⁴4For an arbitrary matrix, this formula does not necessarily define a permutation, only a mapping–which might even map $\left\{1,2,\dots,N\right\}$ to a different set of integers. However, we shall show that in the context we consider it is always a permutation (Remark 7).

With this notation we can explain the relation between the crossing matrices $\mathcal{C}(a)$, $\mathcal{C}(b)$ of two braids and the crossing matrix $\mathcal{C}(ab)$ of their product in the braid group, which we think of as represented by the geometric braid $a$ followed by the geometric braid $b$. The main observation is that the numbering of the strands of $b$ is changed when we premultiply by $a$: the $i^{th}$strand of $b$ becomes an extension of the strand of $a$ which landed at the $i^{th}$ position–that is, in $ab$ it continues the $\bar{\pi}\!\left(i\right)^{th}$ strand. From this it follows that a crossing of the $i^{th}$ strand of $b$ over its $j^{th}$ strand appears in $ab$ as a crossing of the $\bar{\pi}\!\left(i\right)^{th}$ strand over the $\bar{\pi}\!\left(j\right)^{th}$ strand. With this renumbering of strands in $b$, the crossings add, so we have

We will denote this crossing product operation on (crossing) matrices by a circled asterisk:

In [4, §9.1], Thurston defined the order reversal set of a permutation $\pi\in\Sigma_{N}$:

He characterized the sets $S\subset\left\{1,2,\dots,N\right\}\times\left\{1,2,\dots,N\right\}$ which are order reversal sets for some permutation $\pi\in\Sigma_{N}$ via two properties which are easily seen to be necessary, and with a little more work are sufficient:

Thurston then formulated the notion of a permutation braid:

Given a permutation $\pi\in\Sigma_{N}$–which is to say, given the ending position of each strand, we can construct a geometric braid $\pi^{+}$ by joining $(i,1)\in\mathbb{R}^{2}$ to $(\pi\!\left(i\right),0)$ by a straight line segment and making all crossings positive (in case this yields more than two such line segments crossing at the same point, we can perturb a little to get only pairwise crossings). It is clear that no pair of strands crosses more than once. This shows that every $\pi\in\Sigma_{N}$ is the permutation associated to some permutation braid; moreover any permutation braid can be “straightened out” so as to be a $\pi^{+}$. Thus

We caution the reader that this map is not a homomorphism: for example a braid with a single crossing is a permutation braid, but its square is not.

The crossing matrix $\mathbf{R_{\pi}}$ of the permutation braid $\pi^{+}$ is clearly the same as the matrix $R$ with $i,j$ entry $1$ if $(i,j)\in OR\!\left(\pi\right)$ and $0$ otherwise. Since all pairs $(i,j)\in OR\!\left(\pi\right)$ have $i<j$, $R$ is strictly upper triangular ($R_{ij}=0$ for $i\geq j$). The conditions in Proposition 2 characterizing the orientation-reversing set of a permutation can be reinterpreted as conditions on the matrix $R$, which we formulate in

Note that both conditions refer only to entries above the diagonal of $A$, and can be viewed as limitations on the distribution of zero entries (above the diagonal) in $A$.

For any strictly upper triangular $N\times N$ matrix $R$, the positions of its nonzero entries form a set $S$ of pairs $(i,j)$ with $1\leq i<j\leq N$, and Proposition 2 tells us when $S$ is an OR set. The sufficiency argument in the proof of Proposition 2 amounts to saying that, for an $R$-matrix, Equation 2 defines a permutation $\sigma\in\Sigma_{N}$. It is easy to see that, for an $R$-matrix, the term in Equation 2 which is added to $i$ equals the $i^{th}$ row sum

and the subtracted term equals the $i^{th}$ column sum

so Equation 2 really is the same as Equation 1 applied to the $R$-matrix $R$. therefore does not change the permutation defined by Equation 1. Thus, a corollary of Proposition 2 is

To characterize the matrices which arise as crossing matrices (of some, not necessarily positive, braid) we note two necessary conditions:

A braid is $b$ called pure if its permutation $\pi_{b}$ (as defined in §2.3) is the identity permutation. It is easy to see that this condition can be expressed by saying that the $i^{th}$ row sum equals the $i^{th}$ column sum for $i=1,\dots,N$. If for each pair of indices $i,j$ with $1\leq i<j\leq N$ we set $x_{ij}=a_{ij}-a_{ji}$, this condition becomes the system of $N-1$ equations in $\frac{(N-1)(N-2)}{2}$ unknowns of the form $\sum_{j=i+1}^{N}x_{ij}=0$, $i=1,\dots,N-1$. This system has many integer solutions, for example the matrix

has each row sum equal to the corresponding column sum, but if we also throw in the earlier requirement that $a_{ij}-a_{ji}:=x_{ij}\in\left\{0,1\right\}$ we see that the only solution is $x_{ij}=0$ for all $i<j$. Thus

We note that the pure braids form a subgroup $\mathfrak{P}_{N}$ of the braid group $\mathfrak{B}_{N}$, and that the restriction of the crossing matrix mapping to pure braids is a homomorphism to the additive group $\mathfrak{S}_{N}^{0}[\mathbf{Z}]$ of symmetric $N\times N$ integer matrices with zero diagonal. This map is also surjective; to see this, note that each of the symmetric matrices $S_{ij}$ whose only nonzero entries are a “$1$” in the $i,j$ and $j,i$ positions is the crossing matrix of the braid $s_{ij}$ in which strand $i$ crosses over all intermediate strands, “hooks” strand $j$, and then crosses back over all the intermediate strands⁵⁵5 (Note that the latter set of crossings is negative.) to return to its initial position.

Since the $N\times N$ matrices $\mathcal{C}(s_{ij})=S_{ij}$ for $1\leq i<j\leq N$ generate the additive group $\mathfrak{S}_{N}^{0}[\mathbf{Z}]$, this shows

To extend our characterization of crossing matrices to all braids, we note that if $b\in\mathfrak{B}_{N}$ is a braid with permutation $\pi_{b}$, then the braid $s(b):=b(\pi_{b}^{+})^{-1}$ is a pure braid, and so we have a unique factoring $b=s(b)\pi_{b}^{+}$ as a pure braid followed by a permutation braid. It follows that we can write the crossing matrix of $b$ as

where $S=\mathcal{C}(s(b))\in\mathfrak{S}_{N}^{0}[\mathbf{Z}]$ (so $\pi_{S}=id$) and $R=R_{\pi_{b}}$ is an $R$-matrix. Since $R$-matrices are upper triangular, the expression $A=S+R$ is uniquely determined (if it exists) for any matrix; we will refer to it as the $\boldsymbol{SR}$ decomposition of $A$.. We can therefore complete our characterization of crossing matrices for (general) braids:

We denote the set of all $N\times N$ integer matrices with zero diagonal which have an $SR$ decomposition by $\boldsymbol{\mathcal{SR}_{N}}$.

It can sometimes be difficult to immediately visualize the $SR$ decomposition of an integer matrix, so we have adopted a tableau notation to make this clearer: given an $N\times N$ matrix with an $SR$ decomposition $A=S+R$, we exhibit each entry of $A$ $i,j$ strictly above the diagonal ($1\leq i<j\leq N$) as the sum of the corresponding entries $s_{ij}$ of $S$ and $r_{ij}$ of $R$, in the form “$s_{ij}S+r_{ij}R$”; note that

For example, if

then the $\boldsymbol{SR}$ tableau encoding this data is

To complete this picture, we note that the crossing matrix map is not injective. First, we observe that the restriction of the crossing matrix map to the subgroup $\mathfrak{P}_{N}$ is the abelianization of that group; it follows that its kernel is the commutator subgroup of $\mathfrak{P}_{N}$. In general, two braids $a$ and $b$ with the same crossing matrix will have the same permutation (according to the formula in Equation 1) so $c=\bar{a}b$ is a pure braid with zero crossing matrix; it follows that $b=ac$, where $c$ belongs to the commutator subgroup of the pure braid group–that is, $c$ can be written as a product of finitely many braids of the form $[p,q]:=p_{i}q_{i}\bar{p_{i}}\bar{q_{i}}$. We formalize this observation:

It would be natural to expect, in view of Theorem 11, that $\mathcal{C}(\mathfrak{B}_{N}^{+})$ consists of all matrices in $\mathcal{SR}_{N}$ with all entries non-negative. However, this is false; the following are examples of elements of $\mathcal{SR}_{N}$ which, while they have non-negative entries and are crossing matrices of some braids, are not crossing matrices of any positive braids.

It will prove easier to understand these and other examples using their $SR$ tableaux:

and

We will explain why these two matrices are not crossing matrices of any positive braids after developing some properties of such crossing matrices.

The positive braids form a semigroup which includes all permutation braids, so any product of permutation braids, while it may no longer be a permutation braid, will be a positive braid. Conversely, since the standard generators of the braid group are the permutation braids for transpositions of adjacent strands and a positive braid is given by a positive word in these generators, any realization of a matrix $A$ as the crossing matrix of some positive braid can always be expressed as a product of permutation braids, and this corresponds to a factoring of $A$ into a product (with respect to the crossing product operation $\circledast$) of $R$-matrices.

Our problem, then, is how to tell, given a matrix $A$, whether such a factoring is possible. Since any permutation braid is a product of single-crossing braids (i.e., any permutation is a product of transpositions) we can focus on factorization into crossing matrices of transpositions.⁶⁶6 By abuse of notation, we will refer to “factoring into transpositions”.

We note that in addition to the fact that all entries in the crossing matrix of a positive braid are non-negative integers, there is an immediate further restriction on such crossing matrices:

To see the second requirement, note that the argument for $T0$ in the context of $R$-matrices (the necessity of the first condition in Proposition 2) is based only on the assumption that permutation braids are positive. By contrast, the argument for $T1$ in Proposition 2 is based on the assumption that in a permutation braid no pair of strands crosses more than once; this is not necessarily the case for general positive braids.

In view of Remark 14, we define $\boldsymbol{\mathcal{SR}^{+}_{N}}$ to be the set of (non-negative integer, zero diagonal) $T0$ matrices which can be written as the sum of a (non-negative integer) symmetric matrix and an $R$-matrix:

We caution that the $T0$ property for $A$ does not necessarily require $S$, the symmetric part of the decomposition, to be $T0$ on its own (unless, of course, $R=0$, so the braid is pure).

Note that the two examples in Section 3.1 fit this description: $G\in\mathcal{SR}^{+}_{4}$ and $K\in\mathcal{SR}^{+}_{5}$.

To study the factorization problem posed by Remark 14, we formulate a partial inverse to the crossing product operation $\circledast$. If we solve the equation

for $B$ in terms of $A$, we obtain a “division on the left”, defined by

If $A$ and $B$ are to be crossing matrices of positive braids, we must check conditions which insure, for $A\in\mathcal{SR}^{+}_{N}$, that $B\in\mathcal{SR}^{+}_{N}$.

The first requirement is the obvious one, that all entries of the matrix $A-R_{\pi}$ are non-negative; this is the same as requiring that $R_{\pi}\leq A$ entrywise. When this is the case, we will say that the permutation $\pi$ is subordinate to $A$.

The second requirement is that $B$ be $T0$. To understand how this can fail, we take a detour and develop a way to “localize” some of the effects of multiplication ($R_{\pi}\circledast B$) and division ($\pi\backslash A$) on a matrix $B\in\mathcal{SR}^{+}_{N}$ (resp. $A\in\mathcal{SR}^{+}_{N}$).