Characterization of the OU matrix of a braid diagram

In this subsection, we review the definition and some properties of the OU matrix defined in [9]. Let $B$ be an $n$-braid diagram with the strands $s_{1},s_{2},\dots,s_{n}$ that are ordered from the left-hand side to the right-hand side on the top of $B$. Let $\pi$ be a permutation on $(1,2,\dots,n)$. The OU matrix $U(B,\pi)$ of $B$ with $\pi$ is an $n\times n$ zero-diagonal matrix $M$ such that $M(i,j)$ is the number of crossings on the $i^{th}$ strand that are over the $j^{th}$ strand ($i\neq j$), where the “$k^{th}$ strand” stands for the strand $s_{\pi(k)}$, that is, the $k^{th}$ strand according to the order $\pi$. In this paper, we also denote $U(B,id)$ simply by $U(B)$ for the identity permutation $id$. In Figure 2, the OU matrices of a braid diagram $B_{1}$ with the permutations $id$ and $\pi=(4,3,1,2)$ are shown as $U(B_{1})$ and $U(B_{1},\pi)$, respectively.

In this subsection, we define the CN matrix. Let $B$ be an $n$-braid diagram with the strands $s_{1},s_{2},\dots,s_{n}$ ordered from left to right on the top of $B$. A crossing number matrix $N(B)$, or CN matrix, of an $n$-braid diagram $B$ is an $n\times n$ zero-diagonal matrix $M$ such that $M(i,j)$ is the number of crossings between $s_{i}$ and $s_{j}$ ($i\neq j$). By definition, $N(B)$ is a symmetric matrix, and $N(B)=U(B)+\left(U(B)\right)^{T}$ holds for any $B$ (see Figure 2). From Proposition 1, we have the following proposition.

As mentioned in Section 1, the crossing matrix is a matrix defined in [2] for braids. Let $B$ be an $n$-braid diagram with strands $s_{1},s_{2},\dots,s_{n}$ with subscripts from left to right at the top of the braid diagram. The crossing matrix $C(B)$ of $B$ is an $n\times n$ zero-diagonal matrix such that each $(i,j)$ entry is the algebraic number of crossings, i.e., the number of positive crossings minus the number of negative crossings¹¹1The definition of positive/negative crossings in this paper is opposite to [2] and same to [5] or [9]., of $B$ where $s_{i}$ is over $s_{j}$. For a braid diagram $B$, we denote $[B]$ by the equivalent class of a braid that has $B$ as a diagram. We note that if two diagrams $B$ and $B^{\prime}$ represent the same braid in ${\mathbb{R}}^{3}$, i.e., $[B]=[B^{\prime}]$, then $C(B)=C(B^{\prime})$ (see [2, 5]). Therefore, for a braid $b=[B]$ that has a diagram $B$, we can define the crossing matrix $C(b)$ of $b$ by $C(B)$. In [2], the crossing matrices of $n$-braids and pure $n$-braids are completely characterized for all $n\in\mathbb{N}$. More specifically, it is proved in [2] that an $n\times n$ matrix $M$ is the crossing matrix of a pure braid if and only if $M$ is a zero-diagonal integer symmetric matrix. For positive pure braids, it is conjectured as follows.

In this subsection, we discuss the relation between the OU matrix and the crossing matrix. In [9], the following proposition was shown for positive pure braid diagrams.

Let ${\mathcal{B}}_{n}$ be the set of planar isotopy classes of $n$-braid diagrams. Then, ${\mathcal{B}}_{n}$ forms a monoid under the braid product. Let ${\mathcal{P}}_{n}$ be the submonoid of ${\mathcal{B}}_{n}$ which consists of planar isotopy classes of pure braid diagrams and let ${\mathcal{P}}_{n}^{+}$ be the submonoid of ${\mathcal{P}}_{n}$ which consists of planar isotopy classes of positive pure braid diagrams. Let $B_{n}$ be the $n$-braid group, that is, the set of equivalent classes of $n$-braids with the group operation naturally induced by the braid product. Let $P_{n}$ be the subgroup of $B_{n}$ which consists of pure $n$-braids and let $P_{n}^{+}$ be the monoid which consists of positive pure $n$-braids.

Let $X_{n}$ be the set of $n\times n$ zero-diagonal integer matrix and let $X_{n}^{+}$ be the set of matrices $M\in X_{n}$ such that $M(i,j)\geq 0$ for any $i,j$. We note that $(X_{n},+)$ is a group whereas $(X_{n}^{+},+)$ is a monoid. Let $C:B_{n}\to X_{n}$ be the map defined by that $C(b)=C(B)$ is the crossing matrix of $b=[B]\in B_{n}$, let $U:{\mathcal{B}}_{n}\to X_{n}^{+}$ be the map defined by that $U(B)$ is the OU matrix of $B\in{\mathcal{B}}_{n}$ and let $N:{\mathcal{B}}_{n}\to X_{n}^{+}$ be the map defined by $N(B)$ is the CN matrix of $B\in{\mathcal{B}}_{n}$.

For $M\in X_{n}$ and a permutation $\pi\in S_{n}=Sym\{1,\dots,n\}$, let $M^{\pi}$ be the $n\times n$ matrix obtained by rearranging the rows and columns of $M$ according to $\pi$, i.e. the $i^{th}$ row and column are $\pi^{-1}(i)^{th}$ row and column of the original $M$, respectively. We easily see that for $M\in X_{n}$ (resp. $M\in X_{n}^{+}$), it holds that $M^{\pi}\in X_{n}$ (resp. $M^{\pi}\in X_{n}^{+}$), and we obtain a group (resp. monoid) right action of $S_{n}$ on $X_{n}$. Thus, we have semi-direct products $X_{n}\rtimes S_{n}$ (as a group) and $X_{n}^{+}\rtimes S_{n}$ (as a monoid) using the above right action of $S_{n}$ on $X_{n}$ and $X^{+}$.

For any two $n$-braids $a$ and $b\in B_{n}$, it holds that $C(ab)=C(a)+C(b)^{\pi_{a}}$ (see [2, 5]). Thus, the map $C:B_{n}\to X_{n}$ induces a group homomorphism $\widetilde{C}:B_{n}\to X_{n}\rtimes S_{n}$ defined by $\widetilde{C}(b)=(C(b),\rho_{b})$ for $b\in B_{n}$, where $\rho_{b}\in S_{n}$ is the braid permutation of $b$. As noted in [5], ${\rm Ker}\ \widetilde{C}=[P_{n},P_{n}]\times\{id\}$, where $[P_{n},P_{n}]$ is the comutator subgroup of $P_{n}$. On the other hand, for two $n$-braid diagrams $A,B\in{\mathcal{B}}_{n}$, it holds that $U(AB)=U(A)+U(B)^{\pi_{A}}$ (see [9]). Then for each braid diagram $B$ of $b\in B_{n}$ and each $W\in\{C,N\}$, the map $W:{\mathcal{B}}_{n}\to X_{n}^{+}$ induces a monoid homomorphism $\widetilde{W}:{\mathcal{B}}_{n}\to X_{n}^{+}\rtimes S_{n}$ defined by $\widetilde{W}(B)=(W(B),\rho_{b})$. For each $W\in\{C,N\}$, it obviously holds that $\widetilde{W}(B)=(O,id)$ for $B\in{\mathcal{B}}_{n}$, where $O$ is the zero matrix, if and only if $B$ is a “trivial braid diagram”, that is, there are no crossings in $B$.

We say that a matrix $M$ is OU-realizable (resp. CN-realizable) if there exists a braid diagram $B$ such that $U(B)=M$ (resp. $N(B)=M$). We say that a matrix $M$ is C-realizable if there exists a braid $b$ such that $C(b)=M$. By Proposition 5, we have the following proposition.

We have the following proposition from Proposition 3.

In this subsection, we introduce the “BW-ladder diagram” to discuss the CN-realizablity for $(0,2)$-matrices. We call a filled vertex a black vertex and an open vertex a white vertex. A BW-ladder diagram is a ladder-fashioned diagram consisting of some vertical segments and two types of horizontal edges, a black edge and a white edge which have black vertices and white vertices at the endpoints on vertical segments, respectively. In particular, we call a BW-ladder diagram with only black (resp. white) edges a B-ladder diagram (resp. W-ladder diagram). We consider white edges whose endpoints are on vertical segments which are next to each other only. We denote a black edge that has black vertices at the $i^{th}$ and $j^{th}$ vertical segments ($i<j$) from the left-hand side by $B^{i}_{j}$, as shown in Figure 4. Similarly, we denote a white edge that has white vertices at the $i^{th}$ and $(i+1)^{th}$ vertical segments by $W^{i}_{i+1}$. Each BW-ladder diagram can be represented by a word of $B^{i}_{j}$ and $W^{i}_{i+1}$.

For an $n\times n$ strictly upper triangular $(0,2)$-matrix $M$, a B-ladder diagram of $M$ is a B-ladder diagram with $n$ vertical segments that has a black edge $B^{i}_{j}$ if $M(i,j)=2$. Considering $B^{i}_{j}$ as a hook of the two segments where $B^{i}_{j}$ has roots, as shown in the left-hand side in Figure 4, we can restore $M$ from the B-ladder diagram. Moreover, considering $W^{i}_{i+1}$ as a half-twist of the segments where $W^{i}_{i+1}$ has roots, as shown in Figure 4, we consider the following local moves.

Since a W-ladder diagram implies a braid diagram (without crossing information), we have the following proposition.

For $2\times 2$ and $3\times 3$ matrices, we have the following proposition.

In this subsection, we show the following proposition.