Uncrossed Number in Graph Drawing & Knot Theory

• Uncrossed number is a context-dependent invariant that, in graph drawing, measures the minimum number of drawings required so that every edge appears uncrossed in at least one, and in knot theory, it quantifies the minimal crossing changes needed to simplify a knot or link.
• In graph drawing, this invariant is tightly linked with planar embedding parameters such as thickness and outerthickness, with established inequalities offering both lower and upper bounds for various classes of graphs.
• In knot theory, the term encompasses variants like the unlinking number, colored unlinking, and ascending number, each defining the minimal alterations necessary to convert a complex knot or link into its trivial form.

Searching arXiv for the cited papers and related "uncrossed number" usages. Searching "uncrossed number graphs" In recent arXiv literature, “uncrossed number” is not a single standardized invariant. The term is used in several technically distinct settings: as a graph-drawing parameter measuring how many drawings are needed so that every edge is uncrossed at least once; as an interpretation of unlinking or unknotting number in knot and link theory; as a colored unlinking variant that forbids changing inter-component crossings; and, in another knot-theoretic usage, as the ascending number obtained by a traversal-based crossing-change algorithm. Other papers use “uncrossed” in still narrower senses, such as uncrossed gaps in soft random geometric graphs or uncrossed entries in the Sumplete puzzle (Hliněný et al., 2023, Bulai, 2017, DuBois et al., 2019, Davis et al., 2024).

1. Terminological scope and principal meanings

The most systematic modern use of the term occurs in graph drawing. There, an uncrossed collection of drawings of a graph $G$ is a family of drawings such that, for each edge $e\in E(G)$, there is some drawing in which $e$ is uncrossed. The uncrossed number $\mathrm{unc}(G)$ is the minimum number of drawings in such a collection (Hliněný et al., 2023, Balko et al., 2024).

In knot and link theory, the natural interpretation given for “uncrossed number” is the minimum number of crossing changes needed to eliminate all nontrivial knotting or linking. For a link $L$ with $k$ components, this is the unlinking number

$$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$

and for a knot $K$ it is the unknotting number

$$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$

This identification is used explicitly in work on 10-crossing links and on spatial graphs (Bulai, 2017, Buck et al., 2017).

A restricted two-component version appears in colored unlinking, where only self-crossings of the components may be changed and crossings between components are preserved throughout the sequence. In that framework, the “uncrossed number” is the minimal total number of such self-crossing changes needed to unlink (DuBois et al., 2019).

A different knot-theoretic meaning appears in work on the ascending number. There, “uncrossed number” is interpreted as the ascending number $a(K)$: the minimum, over all diagrams, of the number of crossings changed by the roller-coaster algorithm, which converts a diagram into an ascending diagram of the unknot (Davis et al., 2024).

This diversity of meanings suggests that the expression is best treated as context-dependent terminology, not as a universal invariant.

2. Uncrossed number in graph drawing

For a graph $e\in E(G)$0, a drawing places each vertex at a distinct point and each edge as a simple continuous arc between its endpoints, with the standard topological restrictions that edges pass through no other vertices, any two edges share only finitely many points, no two edges touch at an interior point, and no three edges meet at a common interior point. An edge is uncrossed in a drawing if it does not participate in any crossing. A collection $e\in E(G)$1 of drawings is uncrossed if every edge is uncrossed in at least one drawing, and the uncrossed number is

$e\in E(G)$2

This parameter was introduced by Hliněný and Masařík in the GD 2023 line of work and developed further in subsequent papers (Hliněný et al., 2023, Balko et al., 2024).

The graph-drawing uncrossed number sits between classical layering parameters. The established inequalities are

$e\in E(G)$3

where $e\in E(G)$4 is thickness and $e\in E(G)$5 is outerthickness (Balko et al., 2024). The lower bound $e\in E(G)$6 is immediate because the uncrossed edges in each drawing form planar subgraphs that together cover $e\in E(G)$7. The upper bound $e\in E(G)$8 comes from drawing each outerplanar piece in an outerplanar embedding while placing the remaining edges in the outer face (Balko et al., 2024).

A closely related extremal parameter is

$e\in E(G)$9

It yields the basic inequality

$e$0

This observation is central in the later general lower bounds, because bounding $e$1 in a single drawing immediately yields a lower bound on the number of drawings required in an uncrossed collection (Balko et al., 2024, Charvy et al., 28 Jul 2025).

The literature also introduced the uncrossed crossing number, which minimizes the total number of crossings across an uncrossed collection, and the crossing-optimal uncrossed number, the least number of drawings that achieve that minimum. These refinements emphasize that the uncrossed framework is not merely a relaxation of the crossing-number problem; it is a multi-view visualization model in which all edges remain present in every drawing, but each edge must be highlighted uncrossed somewhere (Hliněný et al., 2023).

3. Exact formulas, lower bounds, and complexity in the graph-drawing setting

The first exact formulas were obtained for complete and complete bipartite graphs. For complete graphs,

$e$2

and for complete bipartite graphs with $e$3,

$e$4

These results partly confirm and partly refute the GD 2023 conjecture relating uncrossed number to outerthickness. For complete graphs the conjecture is confirmed, with the only deviation at $e$5, which is planar but not outerplanar. For complete bipartite graphs it is refuted in general: for example, $e$6 while $e$7 (Balko et al., 2024).

For general connected graphs with $e$8 vertices and $e$9 edges, an initial lower bound was

$\mathrm{unc}(G)$0

This bound arises from structural properties of uncrossed subdrawings: they are planar, may be assumed connected, and their facial structure constrains how many additional edges can lie inside their faces (Balko et al., 2024).

A sharper general bound was later proved: $\mathrm{unc}(G)$1 Equivalently, the maximum uncrossed subgraph number satisfies

$\mathrm{unc}(G)$2

In the dense regime $\mathrm{unc}(G)$3, this yields

$\mathrm{unc}(G)$4

For complete graphs, where $\mathrm{unc}(G)$5, the lower bound is asymptotically tight. The same paper gives constructions showing that the $\mathrm{unc}(G)$6-bound is asymptotically tight up to low-order terms for all $\mathrm{unc}(G)$7 in the dense range (Charvy et al., 28 Jul 2025).

The computational complexity picture is also developed. The edge crossing number decision problem—whether a graph admits a drawing with at most $\mathrm{unc}(G)$8 crossed edges—is NP-complete (Balko et al., 2024). The uncrossed crossing number decision problems are NP-complete as well, even when the graph contains an edge $\mathrm{unc}(G)$9 such that $L$0 is planar (Hliněný et al., 2023). By contrast, the uncrossed crossing number is fixed-parameter tractable with parameter equal to the solution size (Hliněný et al., 2023). For the uncrossed number itself, a conditional hardness transfer from Outerthickness is given, while the complexity of Outerthickness remains open (Balko et al., 2024).

4. Uncrossed number as unlinking or unknotting number

In knot theory and low-dimensional topology, “uncrossed number” is naturally interpreted as the minimum number of crossing changes needed to simplify a knot or link to the trivial object. For links, this is the unlinking number; for knots, the unknotting number (Bulai, 2017).

A detailed census appears in the study of prime, non-split 10-crossing links. The universe consists of 287 prime, non-split links with crossing number 10 and at least 2 components. The unlinking number was determined for all but 2 of these links. The unresolved cases are:

• $L$1, for which $L$2, conjectured $L$3;
• $L$4, for which $L$5, conjectured $L$6 (Bulai, 2017).

The paper gives a complete table in Thistlethwaite notation, covering $L$7–$L$8 and $L$9–$k$0, with unlinking numbers ranging from 1 up to 5. This makes clear that crossing number and unlinking number are distinct invariants: in the 10-crossing census all links have $k$1, but $k$2 varies substantially (Bulai, 2017).

The methods are invariant-based. The key tools include the linking-number bound

$k$3

the signature bound

$k$4

the nullity bound

$k$5

and the determinant constraint

$k$6

The paper also uses the cyclic Goeritz presentation obstruction, a signed refinement involving the numbers $k$7 and $k$8 of positive and negative crossing changes,

$k$9

a lattice embedding obstruction derived from Nagel–Owens and Donaldson theory, and Kohn’s covering link method (Bulai, 2017).

Several worked examples illustrate the interaction of lower bounds and explicit crossing-change sequences. For $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$0, decomposing the link into two Hopf-link sublinks with mutual linking number $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$1 gives

$$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$2

and an explicit 5-change sequence shows sharpness. For $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$3, $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$4 yields $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$5, again sharp. For $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$6, the determinant $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$7 rules out $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$8, so $$u(L)=\min\{\text{number of crossing changes needed to convert }L\text{ to the trivial }k\text{-component link}\},$$9. For $K$0, determinant and lattice-embedding obstructions force $K$1, and a 4-change sequence realizes equality (Bulai, 2017).

This knot-theoretic usage is therefore a classical crossing-change minimization invariant, rather than a graph-drawing parameter.

5. Refined knot-theoretic variants: colored unlinking, ascending number, and spatial graphs

A more restrictive framework is colored unlinking for two-component links $K$2 with both components unknotted and linking number zero. Here one allows only crossing changes at self-crossings of the components and never at crossings between components. The component-restricted invariants are

$K$3

and the colored analogue of the classical unlinking number is

$K$4

The basic inequalities are

$K$5

This framework exhibits strong asymmetry: there exist links with small $K$6 but very large $K$7. The paper proves arbitrary asymmetry using a generalized $K$8 family, where $K$9 while $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$0 for the $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$1th member, via branched-cover linking-number calculations (DuBois et al., 2019).

The same paper analyzes all two-component links with at most 10 crossings whose minimal diagrams have linking number zero and both components unknotted. There are 22 such links. Representative values include $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$2 with $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$3, $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$4 with strong asymmetry, and $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$5, where $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$6 but $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$7 (DuBois et al., 2019).

Another interpretation identifies “uncrossed number” with the ascending number $$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$8. Given an oriented knot diagram with a basepoint, the roller-coaster algorithm changes exactly those crossings first encountered from below during a traversal. The ascending number of the diagram is the number of such changes, and

$$u(K)=\min\{\text{number of crossing changes needed to convert }K\text{ to the unknot}\}.$$9

Since the procedure always yields the unknot, one has

$a(K)$0

A principal theorem states that if $a(K)$1 is the closure of a positive braid on $a(K)$2 strands, then

$a(K)$3

Using Rudolph’s formula for positive braid closures,

$a(K)$4

together with a combinatorial lemma $a(K)$5, the paper proves that the roller-coaster algorithm changes exactly $a(K)$6 crossings on the standard positive braid diagram (Davis et al., 2024).

The spatial-graph generalization treats the “uncrossed number” of an embedding $a(K)$7 of a planar graph $a(K)$8 as the minimum number of crossing changes needed to obtain a planar embedding. For such spatial graphs,

$a(K)$9

and the unknotting number is subadditive under order-$e\in E(G)$00 vertex-connected sum: $e\in E(G)$01 For prime $e\in E(G)$02-curves up to seven crossings, the unknotting numbers are determined exactly, with explicit unknotting crossing changes shown for all curves (Buck et al., 2017).

These variants show that, even within knot theory, “uncrossed number” may denote several different crossing-change minimization problems, distinguished by which crossings may be altered and by what counts as the trivial target object.

6. Other specialized uses and sources of ambiguity

Several additional arXiv papers use “uncrossed” in narrower, local, or application-specific senses. In the Sumplete puzzle, uncrossed numbers are simply the entries left uncrossed by the solver; the decision variables $e\in E(G)$03 indicate whether a cell is uncrossed, and the row and column constraints require the sums of uncrossed entries to match the targets. The resulting decision problem is NP-complete even for $e\in E(G)$04-Sumplete, where all entries lie in $e\in E(G)$05 (Ruangwises, 2023).

In one-dimensional soft random geometric graphs, an uncrossed gap is a gap between consecutive points such that no edge joins any point on its left to any point on its right. If $e\in E(G)$06 denotes the number of uncrossed interior gaps, then in the scaling $e\in E(G)$07 the paper proves

$e\in E(G)$08

showing that uncrossed gaps are negligible at the isolated-node threshold for soft RGGs (Wilsher et al., 2020).

In work on the crossing number of $e\in E(G)$09 with an uncrossed Hamiltonian cycle, “uncrossed” refers not to a minimization parameter but to a drawing constraint: the Hamiltonian cycle along the boundary circle must cross no edges. The paper proves that any such drawing has at least

$e\in E(G)$10

with $e\in E(G)$11, and gives an explicit 2-page construction with exactly the Harary–Hill number

$e\in E(G)$12

crossings (Kane, 2013).

In rectilinear drawings of $e\in E(G)$13, the crossing profile $e\in E(G)$14 records how many edges have exactly $e\in E(G)$15 crossings. Here $e\in E(G)$16 is the number of uncrossed edges. For $e\in E(G)$17, the extremal rectilinear values satisfy

$e\in E(G)$18

and the paper characterizes the asymptotic behavior of cumulative quantities $e\in E(G)$19 (Chen et al., 9 Jan 2025).

The overall implication is that “uncrossed number” is best understood as a family resemblance term. In graph drawing it denotes a formal invariant $e\in E(G)$20; in knot theory it usually denotes some variant of unlinking, unknotting, or ascending number; and in other areas it may refer only to uncrossed elements, edges, or gaps in a particular model. Any technical use therefore requires the ambient domain and the exact definition to be specified explicitly.