Braid Charts: Combinatorial Encodings

• Braid charts are combinatorial descriptions that represent braid monodromy and branched coverings using finite, oriented, and labeled graphs on compact surfaces.
• They simplify complex topological constructions by using local vertex models for branch points, commutation, and braid relations.
• The framework extends to surface-links, 3-dimensional braids, and algebraic varieties, providing practical tools for explicit computations and unbraiding operations.

Braid charts are combinatorial descriptions of braid monodromy and branched coverings. In the classical setting, a chart of degree $d$ is a finite oriented labeled graph on a compact oriented surface, most often a disk or sphere, whose local vertex types model simple branch points, commutation, and the braid relation; intersection words obtained from transverse loops recover the monodromy of a 2-dimensional braid. In this sense charts convert the topological problem of describing branched coverings into a combinatorial one in terms of finite labeled graphs, and the formalism extends to surface-links, to 3-dimensional braids via curtains, and, in later algebraic work, to affine and toric chart systems associated with braid varieties and Bott–Samelson varieties (Carter et al., 2013, Carter et al., 2012, Cheah, 4 Jun 2026).

1. Classical definitions and local models

Let $\Sigma^2$ be a compact, oriented surface and fix an integer $d\ge 2$. A chart of degree $d$ is a finite, oriented, labeled graph $\Gamma\subset \Sigma^2$ such that no vertex lies on $\partial \Sigma^2$, every edge is oriented and labeled in $\{1,2,\dots,d-1\}$, and every vertex is one of three types: a black vertex of valency $1$, a crossing of valency $4$, or a white vertex of valency $6$ (Carter et al., 2013).

Vertex type	Valency	Local meaning
Black vertex	1	Simple branch point
Crossing	4	Two transversely intersecting edges with labels $\Sigma^2$0, $\Sigma^2$1
White vertex	6	Local braid relation for labels $\Sigma^2$2

The crossing models commutation when $\Sigma^2$3, and the white vertex models the braid relation

$\Sigma^2$4

When $\Sigma^2$5 has nonempty boundary, one may allow $\Sigma^2$6-valent vertices on $\Sigma^2$7; these are external boundary vertices, the “ends” of free edges (Carter et al., 2013).

Several closely related chart conventions appear in the literature. On a surface diagram $\Sigma^2$8 of an oriented surface-knot $\Sigma^2$9, a chart of degree $d\ge 2$0 is a finite labeled oriented graph $d\ge 2$1 whose vertices have degree $d\ge 2$2, or $d\ge 2$3; the degree-$d\ge 2$4 vertices occur where $d\ge 2$5 crosses a sheet-double curve of $d\ge 2$6 (Nakamura, 2015). In another formulation used for knotted surfaces in $d\ge 2$7, a degree-$d\ge 2$8 braid chart $d\ge 2$9 is an embedded oriented graph whose edges are labeled by

$d$0

with black vertices interpreted as sources or sinks, white vertices as Reidemeister III events, and $d$1-valent crossings recording the order of braid generators (Aranda et al., 5 Oct 2025). This suggests that “b braid chart” is a stable local-combinatorial concept, while the global ambient space and the precise sign convention vary with the application.

2. Encoding braid monodromy

Suppose $d$2 is a $d$3-dimensional braid of degree $d$4, so that $d$5 is a simple branched covering of degree $d$6 with $d$7. For a chart $d$8, let $d$9 be the set of black vertices, choose $\Gamma\subset \Sigma^2$0, and let

$\Gamma\subset \Sigma^2$1

be a loop transverse to $\Gamma\subset \Sigma^2$2. If $\Gamma\subset \Sigma^2$3 meets edges $\Gamma\subset \Sigma^2$4 in order, then each intersection contributes $\Gamma\subset \Sigma^2$5, where $\Gamma\subset \Sigma^2$6 depends on whether $\Gamma\subset \Sigma^2$7 crosses the edge in the positive or negative direction relative to the edge’s arrow. The resulting intersection word is

$\Gamma\subset \Sigma^2$8

and the associated homomorphism

$\Gamma\subset \Sigma^2$9

agrees with the braid monodromy $\partial \Sigma^2$0 of $\partial \Sigma^2$1 (Carter et al., 2013).

The correspondence is complete: for every $\partial \Sigma^2$2-dimensional braid $\partial \Sigma^2$3 there is a chart $\partial \Sigma^2$4 with $\partial \Sigma^2$5, and conversely every chart $\partial \Sigma^2$6 arises from a unique $\partial \Sigma^2$7-dimensional braid $\partial \Sigma^2$8 up to isomorphism (Carter et al., 2013). In the sphere-based language of branched coverings, an unoriented permutation-chart records a homomorphism into $\partial \Sigma^2$9, whereas an oriented braid-chart records a homomorphism into $\{1,2,\dots,d-1\}$0; if a simple $\{1,2,\dots,d-1\}$1-fold covering $\{1,2,\dots,d-1\}$2 admits an embedded lift $\{1,2,\dots,d-1\}$3, one may choose $\{1,2,\dots,d-1\}$4 to be orientable so that its monodromy is the braid monodromy of $\{1,2,\dots,d-1\}$5 (Carter et al., 2012).

The monodromy description also admits a direct cut-and-paste realization. Starting with an $\{1,2,\dots,d-1\}$6-chart in $\{1,2,\dots,d-1\}$7, one takes $\{1,2,\dots,d-1\}$8 copies of $\{1,2,\dots,d-1\}$9, cuts sheet $1$0 along those edges labeled $1$1 for which $1$2 or $1$3, and then glues the cut arcs on sheets $1$4 and $1$5 in reverse orientation. The result is a closed oriented surface $1$6 with a simple branched covering $1$7, and in the braid-chart case the same construction can be viewed as an embedded surface in $1$8 where each labeled edge introduces a half-twist in the $1$9-fiber (Carter et al., 2012).

3. Local moves, equivalence, and simplification

Charts are subject to local modifications that preserve monodromy. In the formulation for $4$0-dimensional braids, the basic moves include creation or annihilation of a $4$1-gonal OVAL NEST, the CII-move for commuting distant labels, the CIII-move for the braid relation, and sliding black vertices past crossings or white vertices when labels permit (Carter et al., 2013). These realize, respectively, local identities such as

$4$2

and

$4$3

Two charts describe isomorphic or equivalent $4$4-dimensional braids precisely when they are related by ambient isotopies together with a finite sequence of these chart moves (Carter et al., 2013).

For charts drawn on surface diagrams of surface-knots, the equivalence relation enlarges to include C-moves and Roseman moves. In this setting $4$5 uniquely presents the $4$6-dimensional braid up to ambient isotopy, C-moves, and Roseman moves (Nakamura, 2015). The same paper develops an additional calculus based on adding $4$7-handles equipped with chart loops. In the repeated-pattern case, a handle carrying $4$8 parallel copies of a braid $4$9 around the cocore and $6$0 copies around the core is denoted

$6$1

and the basic $6$2-handle moves include

$6$3

$6$4

and the handle-slide relation

$6$5

These moves support Euclidean-algorithm-style simplifications of repeated-pattern braids and lead to normal forms involving a single nonzero $6$6-parameter (Nakamura, 2015).

For general charts without branch points, the same handle technology becomes an unbraiding operation. By adding finitely many $6$7-handles of the form $6$8 and $6$9, one can eliminate all chart vertices and loops and reduce to

$\Sigma^2$00

where each $\Sigma^2$01 carries only one chart-loop of type $\Sigma^2$02 or is trivial. The weak unbraiding number $\Sigma^2$03 and the unbraiding number $\Sigma^2$04 quantify the minimal numbers of such handles, and a rough bound is

$\Sigma^2$05

If the chart has at least $\Sigma^2$06 black vertices, one can finish the unbraiding with the standard handles $\Sigma^2$07 plus trivial ones, leaving an unknotted chart consisting only of free edges (Nakamura, 2015). In the degree-$\Sigma^2$08 chart of the $\Sigma^2$09-twist-spun trefoil, a single added handle suffices: $\Sigma^2$10

4. Charts for surface-links and chart-based invariants

An $\Sigma^2$11-chart $\Sigma^2$12 determines a simple surface braid $\Sigma^2$13: over the complement of a neighborhood of $\Sigma^2$14, the projection is a trivial $\Sigma^2$15-strand braid; black vertices become simple branch points; and a Hurwitz arc system from a base point to the black vertices records braid words that prescribe the local gluings. After identifying $\Sigma^2$16 with a tubular neighborhood of a standard $\Sigma^2$17-sphere and capping off, one obtains a closed surface-link

$\Sigma^2$18

Kamada’s theorem implies that every oriented surface-link arises, up to ambient isotopy, as the closure of some simple surface braid and hence from some $\Sigma^2$19-chart (Nagase et al., 2023).

Chart enumeration can therefore be used to organize families of surface-links. For $\Sigma^2$20-charts with exactly two crossings and eight black vertices,

$\Sigma^2$21

Nagase and Shima show that every c-minimal chart is, up to C-move equivalence and conjugation, one of the families $\Sigma^2$22, $\Sigma^2$23, or the exceptional chart $\Sigma^2$24; each has exactly two crossings, eight black vertices, and four Type-$\Sigma^2$25 IO-tangles of labels $\Sigma^2$26 or $\Sigma^2$27 (Nagase et al., 2023).

Chart family	Closure type
$\Sigma^2$28, $\Sigma^2$29	Connected surface-links
$\Sigma^2$30, $\Sigma^2$31, $\Sigma^2$32	Exactly two connected components

To distinguish charts that are not separated by combinatorics alone, the paper uses quandle colorings. For a finite quandle $\Sigma^2$33, a coloring assigns quandle elements to the broken sheets of a generic projection, subject at each double point to

$\Sigma^2$34

and the cardinality $\Sigma^2$35 is an ambient isotopy invariant. For the dihedral quandle $\Sigma^2$36, the exact counts include

$\Sigma^2$37

and

$\Sigma^2$38

Moreover, for fixed $\Sigma^2$39, the two-component charts $\Sigma^2$40 and $\Sigma^2$41 with $\Sigma^2$42 have different coloring numbers when one uses $\Sigma^2$43, so their closures are non-isotopic two-component surface-links (Nagase et al., 2023).

5. Curtains and the 3-dimensional generalization

The $\Sigma^2$44-dimensional analogue of a braid chart is a curtain. A curtain $\Sigma^2$45 is a compact, oriented $\Sigma^2$46-complex with faces labeled in $\Sigma^2$47 such that, after identifying $\Sigma^2$48, the slices

$\Sigma^2$49

are degree-$\Sigma^2$50 charts for all but finitely many $\Sigma^2$51. At non-exceptional levels the chart changes by isotopy; at each exceptional level one performs a single chart move or inserts or deletes a free edge. The union of the slice-wise black vertices together with the inserted or deleted free edges forms the internal boundary

$\Sigma^2$52

which is a link (Carter et al., 2013).

Monodromy is again read by intersections. If a loop $\Sigma^2$53 in $\Sigma^2$54 meets the oriented faces of $\Sigma^2$55 transversely, each intersection contributes $\Sigma^2$56 according to the face orientation, yielding a homomorphism

$\Sigma^2$57

Any homomorphism $\Sigma^2$58 sending each meridian to a conjugate of some $\Sigma^2$59 is realized by a curtain $\Sigma^2$60 with $\Sigma^2$61 and $\Sigma^2$62; in particular every $\Sigma^2$63-dimensional braid is charted by a curtain (Carter et al., 2013).

In the sphere-based formulation, a $\Sigma^2$64-dimensional braid-chart in $\Sigma^2$65 is also called a curtain. Its faces are labeled and oriented, and the induced monodromy maps $\Sigma^2$66 to $\Sigma^2$67. When $\Sigma^2$68, forgetting labels $\Sigma^2$69 leaves a Seifert surface for the branch link, and any simple $\Sigma^2$70-fold branched covering $\Sigma^2$71 admits an embedded lift

$\Sigma^2$72

For arbitrary $\Sigma^2$73, every simple $\Sigma^2$74-fold branched covering admits at least an immersed lift

$\Sigma^2$75

whose singularities are transverse double-point loops disjoint from the branch link (Carter et al., 2012). The standard examples in the paper are the trefoil, whose $\Sigma^2$76-fold branched cover is the lens space $\Sigma^2$77, and the $\Sigma^2$78-colored trefoil, whose $\Sigma^2$79-fold cover is $\Sigma^2$80 again (Carter et al., 2012).

6. Diagrammatic and algebraic descendants

The chart formalism has acquired several later reinterpretations. In the theory of braid varieties, a positive braid word

$\Sigma^2$81

determines the affine variety $\Sigma^2$82, and planar graphs called weaves encode moves between braid words. A simplifying weave $\Sigma^2$83 from $\Sigma^2$84 to the half-twist $\Sigma^2$85 induces an injective map

$\Sigma^2$86

whose images give a cell decomposition of $\Sigma^2$87. For Demazure weaves one obtains maximal toric charts

$\Sigma^2$88

and on each chart the holomorphic $\Sigma^2$89-form has the log-canonical shape

$\Sigma^2$90

so that after setting $\Sigma^2$91 one gets exponential Darboux coordinates (Casals et al., 2020).

A different algebraic usage appears for Bott–Samelson varieties. For a word $\Sigma^2$92, there are $\Sigma^2$93 affine charts $\Sigma^2$94, indexed by signed subexpressions

$\Sigma^2$95

and recent work describes these as a network of affine “braid charts” covering $\Sigma^2$96 and, when the Demazure product is $\Sigma^2$97, the braid variety $\Sigma^2$98. Each chart $\Sigma^2$99 carries a canonical Poisson-compatible cluster seed $d\ge 2$00, fraying procedures access the charts with unsupported letters, and the transition maps $d\ge 2$01 are rational quasi-cluster maps (Cheah, 4 Jun 2026). This suggests a broader use of “braid chart” for coordinate systems whose transition data are controlled by braid-like local transformations.

For knotted surfaces in $d\ge 2$02, braid charts also arise from explicit diagrammatic pipelines. A triplane diagram can be converted to a rainbow diagram, then to a braid movie, and finally to a braid chart in $d\ge 2$03. In this framework a degree-$d\ge 2$04 braid chart is an embedded oriented graph whose edges carry signed labels $d\ge 2$05, with black vertices representing births or deaths of trivial $d\ge 2$06-braids and white vertices representing Reidemeister III moves. The resulting indices satisfy

$d\ge 2$07

and also

$d\ge 2$08

A $d\ge 2$09-dimensional Yamada theorem states that the minimal number of Seifert circles in a clustered triplane diagram equals $d\ge 2$10 (Aranda et al., 5 Oct 2025). In this later setting, braid charts function not only as encodings of monodromy but also as explicit outputs of algorithms connecting triplanes, braid words, and surface-braid presentations.