---
title: Arc Index in Knot Theory
url: https://www.emergentmind.com/topics/arc-index
type: topic
---

# Arc Index in Knot Theory

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 [1010.3005][2007.05711]. 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 [2307.11212][2509.21274][2507.01404].

## 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 [2007.05711]. 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 [1010.3005][2307.11212].

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 [2509.21274].

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 [1010.3005][2402.02717].

## 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 [2007.05711]. An \(n\times n\) grid diagram has \(n\) vertical segments and \(n\) 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 [2007.05711]. Consequently,
\[
\alpha(K)=\min\{n\mid K\text{ admits an }n\times n\text{ grid diagram}\}.
\]

The same combinatorial data can be encoded by a **Cromwell matrix**: an \(n\times n\) \(0\)-\(1\) matrix with exactly two \(1\)'s in each row and each column. Joining the two \(1\)'s in each row horizontally and the two \(1\)'s in each column vertically produces a grid diagram [2007.05711][1010.3005]. 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 [2507.01404].

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 [2402.02717][2407.15859].

## 3. Bounds, characterizations, and relations to other invariants

A standard general upper bound is the Bae–Park inequality
\[
\alpha(L)\le c(L)+2
\]
for any knot or non-split link \(L\), where \(c(L)\) is the minimal crossing number [1806.09719]. For alternating links, Morton–Beltrami’s lower bound implies \(\alpha(L)\ge c(L)+2\), so for nonsplit alternating links one has
\[
\alpha(L)=c(L)+2
\]
[1010.3005]. By contrast, Jin–Park proved that a prime link \(L\) is non-alternating if and only if
\[
\alpha(L)\le c(L),
\]
so the crossing number is an upper bound for arc index in the prime non-alternating case [1010.3005].

Within special families, the invariant can often be computed exactly. For pretzel knots \(K=P(-p,q,r)\) with \(p,q,r\ge2\), \(r\ge q\), and at most one of \(p,q,r\) even, Lee and Jin proved:
\[
\alpha(K)=c(K)\ \text{if }q=2,\qquad
\alpha(K)=c(K)-1\ \text{if }p\ge3,\ q=3,\qquad
\alpha(K)=c(K)-2\ \text{if }p\ge5,\ q=4
\]
[1204.0597]. For adequate links, a more structural formula is available:
\[
\alpha(L)=|s_A D|+|s_B D|=c(L)+2\rho(D),
\]
and the conjectural relation
\[
c(L)+2-\alpha(L)\ge 2g_T(L)
\]
is proved with equality for adequate links [2509.21274].

Arc index also admits contact-topological reformulations. Petkova and Schwartz state that Dynnikov–Prasolov’s theorem gives
\[
\alpha(K)=-\overline{\mathrm{tb}}(K)-\overline{\mathrm{tb}}(m(K)),
\]
and further that if a Legendrian representative \(\Lambda\) satisfies \(\mathrm{tb}(\Lambda)=\overline{\mathrm{tb}}(K)-m\), then the minimal grid size representing \(\Lambda\) is \(\alpha(K)+m\) [2307.11212]. 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 \(D\) of a link \(L\),
\[
\alpha(L)-2 \le c_2(D)+\sum_{n>2}(2n-4)c_n(D),
\]
and hence for an \(n\)-crossing number \(c_n(L)\),
\[
\alpha(L)-2\le (2n-4)c_n(L)\quad (n>2)
\]
[2507.01404]. 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 [2007.05711][2402.02717]. A major milestone was the identification of all prime knots with arc index up to 11 and the tabulation of their minimal arc presentations [1010.3005][1010.2916].

The next stage fixed arc index \(12\). The paper “Prime knots with arc index 12 up to 16 crossings” provides the list of prime knots with arc index \(12\) and crossing number at most \(16\), together with minimal grid diagrams, and states that there are **19,513** such prime knots [2007.05711]. The procedure there is explicit: generate \(12\times12\) Cromwell matrices, identify knot types with Knotscape, remove unknots, links, composite knots, and knots already known to have arc index less than \(12\), and then eliminate duplicates [2007.05711].

For crossing number \(13\), the distribution is completely specified. There are **9,988** prime knots with crossing number \(13\); **4,878** are alternating and have arc index \(15\). Among the non-alternating knots, **49**, **399**, **1,412**, and **3,250** have arc index \(10\), \(11\), \(12\), and \(13\), respectively [2402.02717]. The paper “Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13” gives the minimal grid diagrams for the \(3,250\) knots in the last class [2402.02717].

For crossing number \(14\), there are **46,972** prime knots; **19,536** are alternating and have arc index \(16\). Among the non-alternating knots, **17**, **477**, **3,180**, **8,027**, and **15,735** have arc index \(10\), \(11\), \(12\), \(13\), and \(14\), respectively, and there are none with arc index smaller than \(10\) or larger than \(14\) [2407.15859]. The \(8,027\) knots with arc index \(13\) are exhibited by minimal grid diagrams, while the remaining \(15,735\) are shown to have arc index \(14\) using the lower bound from the Kauffman polynomial [2407.15859].

| Crossing number | Arc-index distribution | Source |
|---|---|---|
| \(13\) | \(49\) at \(10\), \(399\) at \(11\), \(1,412\) at \(12\), \(3,250\) at \(13\), \(4,878\) alternating at \(15\) | [2402.02717] |
| \(14\) | \(17\) at \(10\), \(477\) at \(11\), \(3,180\) at \(12\), \(8,027\) at \(13\), \(15,735\) at \(14\), \(19,536\) alternating at \(16\) | [2407.15859] |

These tabulations provide a dense empirical picture of how \(\alpha(K)\) behaves relative to \(c(K)\). 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 [2007.05711][2407.15859].

## 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 [1704.01787]. For prime nonalternating knots, Jin and Lee also studied when the inequality is strict, identifying cases in which
\[
\alpha(K)=c(K)-1,
\]
and giving minimal grid diagrams of some \(13\)- and \(14\)-crossing examples with that value [1106.2723].

In Legendrian knot theory, Petkova and Schwartz expanded the atlas in standard contact three-space to knots of arc index \(10\) [2307.11212]. Their size-\(10\) grid search yields at most **2,686** Legendrian representatives with maximal Thurston–Bennequin number whose underlying smooth knot types have topological arc index \(10\), and **604** topological knots of arc index \(10\) when orientation and mirroring are taken into account [2307.11212]. Because grids of size \(n\) are the same numerical objects as arc presentations of size \(n\), arc index becomes the organizing complexity threshold for the entire atlas.

The invariant also extends beyond knots and links. For a spatial graph \(G\), 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
\[
\alpha(G)\le c(G)+e+b,
\]
where \(c(G)\) is the minimal crossing number, \(e\) the number of edges, and \(b\) the number of bouquet cut-components [1711.08116]. 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 \(2\)-arcs in graphs together with the index of stabilizer subgroups in alternating or symmetric groups, culminating in a classification of connected \((G,2)\)-arc-transitive graphs of odd order with alternating socle [2105.03880].

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 \(45\)–\(55\) Hz band to the power at the \(50\) 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 [2604.17034].

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 \(S_n\) and on two type-\(B\) analogues in the hyperoctahedral group, obtaining product formulas for the corresponding generating functions [1402.0211]. This usage is unrelated to the open-book invariant of knot theory.

The dominant mathematical meaning of arc index remains the knot-theoretic quantity \(\alpha(K)\), but these parallel usages show that the term is not uniform across fields.

Source: https://www.emergentmind.com/topics/arc-index