Arc Index in Knot Theory

• Arc index is a knot invariant defined as the minimal number of arcs required in an arc presentation, reflecting the knot's geometric complexity.
• It is equivalent to the minimal grid number, establishing a combinatorial method to study knots through grid diagrams and Cromwell matrices.
• Computational tabulations use exhaustive grid diagram generation and matrix enumeration to compare arc index with crossing numbers and Legendrian invariants.

Searching arXiv for the relevant literature on arc index, especially the knot-theoretic invariant and its recent tabulation/bounds. {"query":"arc index knot grid diagram arc presentation site:arxiv.org", "max_results": 10} In knot theory, the arc index of a knot or link is the minimal number of arcs in an arc presentation, equivalently the minimal number of pages in an open-book embedding in which each page contains a single properly embedded arc of the link. Through Cromwell’s correspondence with grid diagrams, arc index is also the minimal grid number, so it can be studied either geometrically in an open book or combinatorially through rectangular and grid diagrams (Jin et al., 2010, Jin et al., 2020). The invariant has become a computationally tractable measure of knot complexity, with extensive tabulations, sharp inequalities against crossing number, and strong connections to Legendrian knot theory, Turaev genus, and multi-crossing numbers (Petkova et al., 2023, Vílchez et al., 25 Sep 2025, Ito, 2 Jul 2025).

1. Definition through open books and arc presentations

An arc presentation embeds a knot $K$ into the union of finitely many vertical half-planes whose common boundary is the $z$-axis, so that each half-plane intersects the knot in a single properly embedded curve; these curves are the arcs of the presentation (Jin et al., 2020). If an arc presentation uses $n$ such arcs, then the arc index is

$$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$

Equivalent formulations appear in both $\mathbb{R}^3$ and $S^3$, using an open book with disk pages and a binding axis or circle (Jin et al., 2010, Petkova et al., 2023).

For links, the same definition applies with $L$ in place of $K$. Cromwell showed that connected sum behaves additively up to the standard correction term,

$\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$

so prime links are the natural basic objects for classification (Vílchez et al., 25 Sep 2025).

This open-book viewpoint makes arc index a geometric complexity measure: it asks how efficiently a knot can be routed through a particularly simple fibration-like structure. The data from later tabulations suggest that this efficiency can differ sharply between alternating and non-alternating knots (Jin et al., 2010, Lee et al., 2024).

2. Grid diagrams, Cromwell matrices, and equivalent models

A grid diagram is a knot diagram composed of finitely many horizontal segments and the same number of vertical segments, with the convention that vertical segments always cross over horizontal segments (Jin et al., 2020). An $n\times n$ grid diagram has $z$0 vertical segments and $z$1 horizontal segments, and the paper “Prime knots with arc index 12 up to 16 crossings” states that a grid diagram can be converted easily to an arc presentation with the number of arcs equal to the number of vertical line segments, and conversely an arc presentation can be converted to a grid diagram with the number of vertical line segments equal to the number of arcs (Jin et al., 2020). Consequently,

$z$2

The same combinatorial data can be encoded by a Cromwell matrix: an $z$3 $z$4-$z$5 matrix with exactly two $z$6's in each row and each column. Joining the two $z$7's in each row horizontally and the two $z$8's in each column vertically produces a grid diagram (Jin et al., 2020, Jin et al., 2010). This encoding is central in exhaustive searches, because the geometric problem of minimizing the number of pages becomes a finite enumeration problem over matrices.

Recent work extends the same equivalence to loose rectangular diagrams, in which each vertical arc contains only over-crossings or only under-crossings. A loose rectangular diagram can be converted to a usual arc presentation without increasing the number of vertical segments, so arc index can also be computed by minimizing over loose rectangular diagrams (Ito, 2 Jul 2025).

This combinatorial equivalence is the main reason arc index is algorithmically accessible. It allows the invariant to be attacked with DT codes, spanning-tree constructions, Cromwell matrices, and software such as Knotscape rather than only with three-dimensional embeddings (Lee et al., 2024, Jin et al., 2024).

3. Bounds, characterizations, and relations to other invariants

A standard general upper bound is the Bae–Park inequality

$z$9

for any knot or non-split link $n$0, where $n$1 is the minimal crossing number (No et al., 2018). For alternating links, Morton–Beltrami’s lower bound implies $n$2, so for nonsplit alternating links one has

$n$3

(Jin et al., 2010). By contrast, Jin–Park proved that a prime link $n$4 is non-alternating if and only if

$n$5

so the crossing number is an upper bound for arc index in the prime non-alternating case (Jin et al., 2010).

Within special families, the invariant can often be computed exactly. For pretzel knots $n$6 with $n$7, $n$8, and at most one of $n$9 even, Lee and Jin proved: $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$0 (Lee et al., 2012). For adequate links, a more structural formula is available: $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$1 and the conjectural relation

$$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$2

is proved with equality for adequate links (Vílchez et al., 25 Sep 2025).

Arc index also admits contact-topological reformulations. Petkova and Schwartz state that Dynnikov–Prasolov’s theorem gives

$$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$3

and further that if a Legendrian representative $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$4 satisfies $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$5, then the minimal grid size representing $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$6 is $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$7 (Petkova et al., 2023). This suggests that arc index is not merely a diagrammatic invariant; it also controls the vertical structure of Legendrian mountain ranges.

A recent multi-crossing generalization shows that for a non-split multi-crossing diagram $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$8 of a link $$\alpha(K)=\min\{n\mid K\text{ admits an arc presentation with }n\text{ arcs}\}.$$9,

$\mathbb{R}^3$0

and hence for an $\mathbb{R}^3$1-crossing number $\mathbb{R}^3$2,

$\mathbb{R}^3$3

(Ito, 2 Jul 2025). This subsumes several earlier inequalities for triple and quadruple crossing numbers.

4. Computational tabulation and census results

The computational study of arc index proceeds by exhaustive generation of grid diagrams or Cromwell matrices, knot identification with Knotscape, and elimination of duplicates and knots already known to have smaller arc index (Jin et al., 2020, Lee et al., 2024). A major milestone was the identification of all prime knots with arc index up to 11 and the tabulation of their minimal arc presentations (Jin et al., 2010, Jin et al., 2010).

The next stage fixed arc index $\mathbb{R}^3$4. The paper “Prime knots with arc index 12 up to 16 crossings” provides the list of prime knots with arc index $\mathbb{R}^3$5 and crossing number at most $\mathbb{R}^3$6, together with minimal grid diagrams, and states that there are 19,513 such prime knots (Jin et al., 2020). The procedure there is explicit: generate $\mathbb{R}^3$7 Cromwell matrices, identify knot types with Knotscape, remove unknots, links, composite knots, and knots already known to have arc index less than $\mathbb{R}^3$8, and then eliminate duplicates (Jin et al., 2020).

For crossing number $\mathbb{R}^3$9, the distribution is completely specified. There are 9,988 prime knots with crossing number $S^3$0; 4,878 are alternating and have arc index $S^3$1. Among the non-alternating knots, 49, 399, 1,412, and 3,250 have arc index $S^3$2, $S^3$3, $S^3$4, and $S^3$5, respectively (Lee et al., 2024). The paper “Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13” gives the minimal grid diagrams for the $S^3$6 knots in the last class (Lee et al., 2024).

For crossing number $S^3$7, there are 46,972 prime knots; 19,536 are alternating and have arc index $S^3$8. Among the non-alternating knots, 17, 477, 3,180, 8,027, and 15,735 have arc index $S^3$9, $L$0, $L$1, $L$2, and $L$3, respectively, and there are none with arc index smaller than $L$4 or larger than $L$5 (Jin et al., 2024). The $L$6 knots with arc index $L$7 are exhibited by minimal grid diagrams, while the remaining $L$8 are shown to have arc index $L$9 using the lower bound from the Kauffman polynomial (Jin et al., 2024).

Crossing number	Arc-index distribution	Source
$K$0	$K$1 at $K$2, $K$3 at $K$4, $K$5 at $K$6, $K$7 at $K$8, $K$9 alternating at $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$0	(Lee et al., 2024)
$\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$1	$\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$2 at $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$3, $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$4 at $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$5, $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$6 at $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$7, $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$8 at $\alpha(L_1\#L_2)=\alpha(L_1)+\alpha(L_2)-2,$9, $n\times n$0 at $n\times n$1, $n\times n$2 alternating at $n\times n$3	(Jin et al., 2024)

These tabulations provide a dense empirical picture of how $n\times n$4 behaves relative to $n\times n$5. A plausible implication is that the arc-index census now functions both as a database for testing conjectures and as a benchmark suite for algorithms on grid diagrams, contact invariants, and knot recognition (Jin et al., 2020, Jin et al., 2024).

5. Extensions to special families, Legendrian knots, and spatial graphs

Arc index has been computed or controlled in several specialized settings beyond census work. For some infinite families of mutant Montesinos knots, Jin and Lee found that mutation does not change the arc index; they constructed infinitely many mutant pairs and triples of nonalternating Montesinos knots with the same arc index (Jin et al., 2017). For prime nonalternating knots, Jin and Lee also studied when the inequality is strict, identifying cases in which

$n\times n$6

and giving minimal grid diagrams of some $n\times n$7- and $n\times n$8-crossing examples with that value (Jin et al., 2011).

In Legendrian knot theory, Petkova and Schwartz expanded the atlas in standard contact three-space to knots of arc index $n\times n$9 (Petkova et al., 2023). Their size-$z$00 grid search yields at most 2,686 Legendrian representatives with maximal Thurston–Bennequin number whose underlying smooth knot types have topological arc index $z$01, and 604 topological knots of arc index $z$02 when orientation and mirroring are taken into account (Petkova et al., 2023). Because grids of size $z$03 are the same numerical objects as arc presentations of size $z$04, arc index becomes the organizing complexity threshold for the entire atlas.

The invariant also extends beyond knots and links. For a spatial graph $z$05, Lee, No, and Oh defined arc presentations in an open book where all vertices lie on the binding axis and proved the sharp upper bound

$z$06

where $z$07 is the minimal crossing number, $z$08 the number of edges, and $z$09 the number of bouquet cut-components (Lee et al., 2017). This generalizes the classical link inequality and suggests that arc index is robust under passage from one-dimensional knots to embedded graph structures.

6. Other meanings of the term

Although arc index is standard in knot theory, the phrase is discipline-dependent. In graph theory, the paper “Two-arc-transitive graphs of odd order -- II” does not define arc index as a standalone invariant; instead it studies arcs and $z$10-arcs in graphs together with the index of stabilizer subgroups in alternating or symmetric groups, culminating in a classification of connected $z$11-arc-transitive graphs of odd order with alternating socle (Li et al., 2021).

In electric arc welding, “arc index” refers to the Arc Stability Index (ASI), defined from the short-time Fourier transform of the welding current as a ratio of spectral energy in the $z$12–$z$13 Hz band to the power at the $z$14 Hz fundamental. In that setting, ASI is an energy-based diagnostic feature for classifying transient, stable, and extinction regimes, and it is integrated with spectral entropy and harmonic distortion in an STFT–ML pipeline (Akinci et al., 18 Apr 2026).

In algebraic combinatorics, the phrase appears indirectly through arc permutations. Elizalde and Roichman studied descent set, major index, and flag-major index on arc permutations in $z$15 and on two type-$z$16 analogues in the hyperoctahedral group, obtaining product formulas for the corresponding generating functions (Elizalde et al., 2014). This usage is unrelated to the open-book invariant of knot theory.

The dominant mathematical meaning of arc index remains the knot-theoretic quantity $z$17, but these parallel usages show that the term is not uniform across fields.