Crossing Matrix of a Braid

• Crossing matrix of a braid is an integral model encoding the signed overcrossings of strands, remaining invariant under braid group relations.
• It decomposes into a symmetric zero-diagonal pure component and an upper-triangular permutation part, facilitating classification of braid invariants.
• Its functorial properties underpin advances in quantum representations, link invariants, and algorithmic braid recognition.

A crossing matrix encodes the signed intersection pattern of strands in a braid; it provides an integral matrix model that is functorial with respect to braid composition and robust under braid group relations. The crossing matrix is a key algebraic and combinatorial invariant in the structure theory of braid groups, with applications ranging from braid group representations to link invariants, and, more recently, surface braid systems and quantum information.

1. Definition and Construction

Let $b \in \mathfrak B_n$ be a braid in the $n$-strand Artin braid group, presented diagrammatically with strands labeled $1$ to $n$ at the top and bottom. The crossing matrix $C(b)$ is the $n \times n$ integer matrix whose off-diagonal entries

$C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$

record the algebraic number of signed overcrossings from strand $i$ to strand $j$. The diagonal entries vanish: $C(b)_{ii} = 0$. This matrix is independent of the chosen diagram due to invariance under the Artin braid relations $n$0, $n$1 $n$2, and $n$3 (Shimizu, 10 Sep 2025, Shimizu et al., 5 Jan 2026, Kuno et al., 25 Nov 2025).

Given a braid word $n$4, $n$5 is constructed by sequentially inserting, for each letter, a $n$6 (resp. $n$7) in the $n$8 (resp. $n$9) position for $1$0 (resp. $1$1), under the current permutation of strands induced so far in the word (Kuno et al., 25 Nov 2025).

2. Algebraic Properties and Functoriality

The crossing matrix induces a crossed homomorphism $1$2, reflecting the semidirect product structure of $1$3 with the natural $1$4-action: $1$5 where $1$6 is the permutation of $1$7 induced by $1$8, acting on matrices by simultaneous permutation of rows and columns. Under conjugation,

$1$9

where $n$0 acts as a permutation matrix on the indices (Kuno et al., 25 Nov 2025, Gutierrez et al., 2018).

When restricted to the pure braid group $n$1, $n$2 is a genuine homomorphism to symmetric zero-diagonal integer matrices, and provides the abelianization of $n$3, with kernel $n$4 (Gutierrez et al., 2018).

3. Characterization and Classification

For general braids, every crossing matrix decomposes as $n$5 where $n$6 is symmetric zero-diagonal (from the pure component), and $n$7 is a strictly upper-triangular 0–1 matrix associated with permutation braids, subject to Thurston’s T0 and T1 axioms:

• (T0): $n$8 and $n$9 $C(b)$0 for all $C(b)$1
• (T1): $C(b)$2 $C(b)$3

A matrix $C(b)$4 is the crossing matrix of some $C(b)$5 if and only if such a decomposition exists (Gutierrez et al., 2018).

For positive pure braids, $C(b)$6 is nonnegative, symmetric, and satisfies T0. For $C(b)$7, such matrices coincide with realizable crossing matrices of positive pure braids. The T0 condition reflects transitive-exclusion: if strands $C(b)$8 and $C(b)$9, and $n \times n$0 and $n \times n$1 do not cross, then neither can $n \times n$2 and $n \times n$3 (Ozawa et al., 10 Jun 2025, Shimizu et al., 22 Feb 2025).

4. Invariants Derived from the Crossing Matrix

Several braid invariants are constructed from $n \times n$4:

• Exponent sum ("writhe"): $n \times n$5
• Purified determinant $n \times n$6: for $n \times n$7 with induced permutation of order $n \times n$8, $n \times n$9, which is a conjugacy invariant and satisfies $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$0 (Shimizu, 10 Sep 2025)
• Characteristic polynomial: $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$1, yielding a multiset of eigenvalues; invariance holds under conjugation, and its essential eigenvalues are stable under Hurwitz moves in braid systems (Shimizu et al., 5 Jan 2026)
• Johnson homomorphism: $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$2, when suitably mapped, yields the extended first Johnson homomorphism $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$3 (Kuno et al., 25 Nov 2025).

5. Connections to Braid Group Representations

Path-analyzing and quantum representations encode the combinatorics of crossings via generalized crossing matrices:

• In Pourkia’s construction, matrices with entries in $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$4 capture the path history of each strand; the unreduced Burau representation is obtained for suitable parameter specialization (Pourkia, 2018).
• In quantum topology, each over-crossing is replaced by a Yang–Baxter $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$5-matrix acting on tensor powers of $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$6; the full “crossing matrix” is the ordered product of these operators, yielding a unitary representation of the braid group (Ben-Aryeh, 2014).

6. Applications: Braid and Link Invariants, Surface Braids, and Decision Problems

Crossing matrices play a pivotal role in:

• Algorithmic braid recognition: Algorithms exist to construct all positive braid words yielding a given matrix or detect non-realizability via T0 violations (Gutierrez et al., 2018)
• Surface braids and surface links: Crossing matrices, via their characteristic polynomials, produce invariants for braid systems up to Hurwitz equivalence; essential eigenvalues detect the necessity of Euler fusion/fission steps in surface link equivalence (Shimizu et al., 5 Jan 2026)
• Link theory: While the determinant $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$7 generally does not descend to a link invariant due to non-invariance under Markov moves, the construction suggests enhancements for link invariance (Shimizu, 10 Sep 2025)

7. Realizability and Structural Characterization

Matrix models such as the CN matrix and OU matrix are derived from or related to the crossing matrix:

• CN matrix: For a braid projection, the symmetric, zero-diagonal matrix of total crossings between strand pairs, characterized for pure 6-braids as even, nonnegative, T0 matrices (Ozawa et al., 10 Jun 2025)
• OU matrix: Number of over-crossings (ignoring sign), with characterization via symmetrization and T0 (Shimizu et al., 22 Feb 2025)
• Positive braids: Crossing matrices further constrained to be nonnegative and admit a recursive decomposition into contribution of elementary generators, yielding an effective realization algorithm (Gutierrez et al., 2018)

These frameworks provide necessary and sufficient conditions for the existence of a braid diagram corresponding to a given crossing matrix for pure and positive braids up to $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$8; for larger $C(b)_{ij} = \bigl(\#\text{positive crossings where %%%%6%%%% passes over %%%%7%%%%}\bigr) - \bigl(\#\text{negative crossings where %%%%8%%%% passes over %%%%9%%%%}\bigr)$9, the T0-based criteria and their generalizations remain conjectural.

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References:

Key foundational results and recent advances on the crossing matrix and related invariants are found in (Shimizu, 10 Sep 2025, Kuno et al., 25 Nov 2025, Shimizu et al., 5 Jan 2026, Ozawa et al., 10 Jun 2025, Shimizu et al., 22 Feb 2025), and (Gutierrez et al., 2018). For connections to quantum representations, see (Pourkia, 2018) and (Ben-Aryeh, 2014).