Tropical Crossing Number

• Tropical crossing number is a planar complexity invariant defined as the minimum weighted count of nodes in tropical immersions, encapsulating both metric and combinatorial properties.
• It employs dual Newton polygon subdivisions to translate geometric features of tropical plane curves into precise nodal multiplicities.
• The invariant is contrasted with classical crossing numbers, highlighting differences due to metric realizations and guiding faithful tropicalizations in algebraic geometry.

Searching arXiv for recent and foundational papers on tropical crossing number. Tropical crossing number is a planar complexity invariant for tropical curves and graphs. In the metric setting introduced for abstract tropical curves, it is the minimum number of self-intersections of a planar tropical immersion, with each node counted with its tropical multiplicity (Cartwright et al., 2014). In a later extension to finite, non-metric graphs, the invariant is defined by minimizing over all choices of edge lengths on the graph, so that one asks for the fewest tropical nodes required by any metric realization of the underlying combinatorial graph (Cape et al., 19 Aug 2025). A related but distinct line of work shows that for metric graphs realized as balanced tropical curves equipped with the lattice length metric, the relevant planar crossing count can coincide exactly with the classical graph crossing number (Campo et al., 2016). Together these works place tropical crossing number at the interface of tropical plane geometry, metric graph theory, Newton polygon combinatorics, and realizability problems.

1. Definition and basic framework

The foundational definition appears in the study of planar immersions of abstract tropical curves. There, an abstract tropical curve is a finite connected graph, possibly with loops or multiple edges, whose edges are assigned positive real lengths or infinity, with the convention that any infinite edge has a 1-valent endpoint (Cartwright et al., 2014). Two such curves are considered equivalent up to subdivisions and reverse subdivisions (Cartwright et al., 2014).

An embedding of an abstract tropical curve in $\mathbb{R}^n$ is a smooth tropical curve in $\mathbb{R}^n$ that is isometric to a tropical modification of the given curve, while an immersion in $\mathbb{R}^2$ is the analogous notion with a nodal tropical curve in the plane (Cartwright et al., 2014). For a trivalent abstract tropical curve $\Gamma$, the tropical crossing number is defined as the minimum number of nodes, counted with multiplicity, among all planar immersions of $\Gamma$ (Cartwright et al., 2014). Equivalently, it is the minimum total node multiplicity over all nodal tropical curves in $\mathbb{R}^2$ that are isometric to a tropical modification of $\Gamma$ (Cartwright et al., 2014).

The multiplicity at a node is intrinsic to the tropical structure. If the two primitive integral direction vectors are $v=(v_1,v_2)$ and $u=(u_1,u_2)$, then the node multiplicity is

$$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$

This agrees with the stable intersection multiplicity of the two tropical lines (Cartwright et al., 2014). Consequently, a tropical crossing is not merely a geometric transverse intersection in a graph drawing; it is a node weighted by local integral data.

A finite-graph version was introduced later. For a finite, non-metric graph $\mathbb{R}^n$0, one defines

$\mathbb{R}^n$1

where the minimum is taken over all abstract tropical curves $\mathbb{R}^n$2 whose underlying finite skeleton is $\mathbb{R}^n$3 (Cape et al., 19 Aug 2025). This separates the dependence on metric data from the combinatorial graph itself. In this setting, $\mathbb{R}^n$4 if and only if $\mathbb{R}^n$5 is tropically planar, meaning it is the skeleton of some smooth tropical plane curve (Cape et al., 19 Aug 2025).

2. Tropical plane curves, nodes, and Newton polygons

The invariant is naturally expressed through planar tropical curves and their dual subdivisions. A tropical plane curve is defined by a tropical polynomial

$\mathbb{R}^n$6

where tropical addition is max and tropical multiplication is ordinary addition (Cape et al., 19 Aug 2025). By the Structure Theorem, such a curve is a balanced 1-dimensional polyhedral complex (Cape et al., 19 Aug 2025).

Its Newton polygon is the convex hull of the exponent vectors of the monomials of $\mathbb{R}^n$7, and a regular subdivision of the Newton polygon is obtained from a height function and the upper convex hull construction (Cape et al., 19 Aug 2025). Duality governs the geometry: a subpolygon corresponds to a vertex of the tropical curve, shared edges of subpolygons correspond to edges of the curve, and boundary edges correspond to rays (Cape et al., 19 Aug 2025).

A smooth tropical plane curve corresponds to a unimodular triangulation of the Newton polygon, and its skeleton is trivalent (Cape et al., 19 Aug 2025). A nodal subdivision allows, in addition to unimodular triangles, unit parallelograms of area $\mathbb{R}^n$8; each such parallelogram is dual to a 4-valent node in the tropical curve (Cape et al., 19 Aug 2025). This supplies a direct combinatorial model for crossings: triangles correspond to ordinary trivalent pieces, while parallelograms correspond to nodes.

The genus formula used repeatedly in this literature states that if a connected skeleton arises from a nodal subdivision with $\mathbb{R}^n$9 interior lattice points and $\mathbb{R}^2$0 nodes, then

$\mathbb{R}^2$1

This relation provides the bridge between the combinatorics of a Newton polygon and the genus of the resulting graph (Cape et al., 19 Aug 2025). In the metric-graph setting, a closely related statement appears as the identity $\mathbb{R}^2$2 for a planar immersion of genus $\mathbb{R}^2$3 with $\mathbb{R}^2$4 nodes counted with multiplicity, where $\mathbb{R}^2$5 is the number of integral points in the interior of the dual Newton polygon (Cartwright et al., 2014). The two formulations reflect different placements of the same parameters in dual constructions.

3. Relation to classical crossing number

A central issue is how tropical crossing number compares with the ordinary graph-theoretic crossing number. In the original metric theory, the tropical crossing number is always bounded below by the usual graph-theoretic crossing number of the underlying graph (Cartwright et al., 2014). However, the two invariants can differ dramatically because the tropical invariant depends on the metric and on node multiplicities.

This distinction is illustrated by the family of chains of loops with bridges. These graphs are planar as abstract graphs, yet for generic edge lengths their tropical crossing number can still be quadratic (Cartwright et al., 2014). This shows that graph planarity does not imply tropical planarity for a prescribed metric realization.

At the same time, another line of work establishes a strong equivalence in a different framework. For a metric graph $\mathbb{R}^2$6, the classical crossing number $\mathbb{R}^2$7 is the minimum number of edge intersections among all embeddings of $\mathbb{R}^2$8 into $\mathbb{R}^2$9 (Campo et al., 2016). The main theorem of "Tropical Embeddings of Metric Graphs" proves that every abstract metric graph $\Gamma$0 admits an isometric balanced embedding in the plane, after a tropical modification, with exactly $\Gamma$1 crossings restricted to the embedded edges (Campo et al., 2016). In the language used there, the relevant planar tropical crossing count can therefore be taken to be the same classical invariant: $\Gamma$2 in the setting they construct (Campo et al., 2016).

This does not erase the distinction drawn in the metric-immersion literature. Rather, it shows that under the specific notion of balanced tropical realization with lattice length metric, the planar tropical obstruction can coincide exactly with the classical obstruction (Campo et al., 2016). A plausible implication is that “tropical crossing number” has been used in closely related but not identical senses, depending on whether one works with nodal immersions of abstract tropical curves or with balanced tropical embeddings of metric graphs.

4. Structural results and bounds

The main structural theorem of the immersion-based theory states that every abstract tropical curve embeds in sufficiently high-dimensional Euclidean space, and every trivalent abstract tropical curve immerses in the plane (Cartwright et al., 2014). More precisely, if the maximum vertex degree is $\Gamma$3, then the curve has a smooth embedding in $\Gamma$4 for

$\Gamma$5

and a planar immersion when $\Gamma$6 (Cartwright et al., 2014).

For trivalent curves with $\Gamma$7 edges, the tropical crossing number is at most $\Gamma$8, and this bound is sharp up to constants (Cartwright et al., 2014). The construction is combinatorial: after subdividing loops and multiple edges, the graph is drawn piecewise linearly using directions among

$\Gamma$9

with generic perturbations ensuring that in the planar case self-intersections occur only as isolated nodes (Cartwright et al., 2014). Because the planar construction effectively breaks each original edge into four segments and uses only directions among $\Gamma$0, any pairwise crossing multiplicity is at most one (Cartwright et al., 2014).

Lower bounds are obtained via the Newton polygon and divisorial gonality. If $\Gamma$1 has divisorial gonality $\Gamma$2 and genus $\Gamma$3, then

$\Gamma$4

The proof uses the lattice width of the dual Newton polygon and area bounds derived from polygon theory, combined with Pick’s theorem (Cartwright et al., 2014).

The finite-graph extension imports these bounds and reinterprets them combinatorially. If $\Gamma$5 is a trivalent graph of genus $\Gamma$6, then

$\Gamma$7

because a trivalent genus-$\Gamma$8 graph has

$\Gamma$9

and the metric-graph bound gives $\mathbb{R}^2$0 (Cape et al., 19 Aug 2025). More significantly, there exists a family of graphs whose tropical crossing number grows quadratically in the genus (Cape et al., 19 Aug 2025). The lower bound comes from

$\mathbb{R}^2$1

together with

$\mathbb{R}^2$2

which yields

$\mathbb{R}^2$3

For expander graphs, treewidth grows linearly in the number of vertices, so the resulting lower bound is quadratic in genus (Cape et al., 19 Aug 2025).

The authors note, however, that this method does not currently produce a planar family with quadratic growth; that remains open (Cape et al., 19 Aug 2025).

5. Exact computations and representative examples

The literature contains several exact computations in low genus and in finite-graph settings. For genus $\mathbb{R}^2$4, the crossing number is $\mathbb{R}^2$5 exactly for trees whose underlying graph is a subdivision of either a caterpillar graph or a windmill graph (Cartwright et al., 2014). For genus $\mathbb{R}^2$6, the paper studies sun graphs, a cycle with $\mathbb{R}^2$7 rays attached. If $\mathbb{R}^2$8, then

$\mathbb{R}^2$9

and this bound is sharp for suitable choices of edge lengths; for $\Gamma$0 there are planar embeddings (Cartwright et al., 2014).

For stable genus-two tropical curves, there are only two combinatorial types: the barbell graph and the theta graph (Cartwright et al., 2014). The barbell graph always has crossing number $\Gamma$1 (Cartwright et al., 2014). For the theta graph with edge lengths $\Gamma$2, one has a planar embedding if $\Gamma$3, also if $\Gamma$4, but if $\Gamma$5, then the tropical crossing number is $\Gamma$6 (Cartwright et al., 2014). The equilateral theta graph is therefore the only nonplanar stable genus-two case in that classification.

The finite-graph theory supplies exact values beyond the metric setting. Its first main theorem states that for any integers $\Gamma$7, there exists a graph $\Gamma$8 such that

$\Gamma$9

The construction uses a non-planar graph $v=(v_1,v_2)$0 with $v=(v_1,v_2)$1 and a crowded graph $v=(v_1,v_2)$2 with $v=(v_1,v_2)$3, then stitches copies together using bridges (Cape et al., 19 Aug 2025). This shows that every nonnegative integer occurs as a tropical crossing number for a finite graph and that the classical crossing number can be prescribed independently up to that value (Cape et al., 19 Aug 2025).

The smallest non-tropically planar graph addressed explicitly is the genus-3 lollipop graph. The exact result is

$v=(v_1,v_2)$4

The proof is computational: realizations with one node from genus-$v=(v_1,v_2)$5 polygons and with two nodes from genus-$v=(v_1,v_2)$6 polygons are ruled out by enumerating regular subdivisions, and an explicit nodal subdivision with three unit parallelograms is exhibited (Cape et al., 19 Aug 2025). The paper presents this as the first exact computation of the tropical crossing number of the genus-3 lollipop graph (Cape et al., 19 Aug 2025).

6. Methods of proof and computation

Two complementary methodological traditions appear in the literature. The first is constructive and geometric. In the planar embedding theorem for metric graphs, one begins with a minimal-crossing embedding in $v=(v_1,v_2)$7 with rational slopes and then adjusts lengths without changing crossings by inserting local gadgets called créneaux (Campo et al., 2016). A créneau is a zig-zag path with balancing rays added so that the tropical balancing condition holds, allowing the tropical length of an edge to be increased to any desired value while staying inside a tiny neighborhood of the original segment and hence introducing no new crossings (Campo et al., 2016). This yields an isometric balanced embedding with exactly $v=(v_1,v_2)$8 crossings (Campo et al., 2016).

The second tradition is combinatorial and computational. In the immersion-based theory, lower bounds come from dual Newton polygons, Pick’s theorem, polygonal width estimates, and gonality arguments (Cartwright et al., 2014). For finite graphs, the lollipop computation proceeds by exhaustive enumeration of regular subdivisions of maximal lattice polygons. The procedure is explicitly organized as follows: enumerate maximal lattice polygons of genus $v=(v_1,v_2)$9 up to $u=(u_1,u_2)$0 plus translation, enumerate all regular unimodular triangulations, check whether the desired skeleton occurs, and if not, move to polygons of genus $u=(u_1,u_2)$1 allowing $u=(u_1,u_2)$2 unit parallelograms until the graph is found (Cape et al., 19 Aug 2025).

A crucial reduction in that search is the restriction to non-hyperelliptic polygons for non-chain graphs. The relevant proposition states that if $u=(u_1,u_2)$3 is a hyperelliptic Newton polygon of genus $u=(u_1,u_2)$4, and $u=(u_1,u_2)$5 is the skeleton of a nodal subdivision of $u=(u_1,u_2)$6 with at most $u=(u_1,u_2)$7 unit parallelograms, then $u=(u_1,u_2)$8 is a chain (Cape et al., 19 Aug 2025). Since the lollipop graph is not a chain, hyperelliptic polygons can be discarded in that computation (Cape et al., 19 Aug 2025).

The finite-graph paper also proves a patching regular subdivisions proposition: if $u=(u_1,u_2)$9 are Newton polygons with regular subdivisions $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$0, and they are glued along an edge of lattice length $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$1 to form a convex polygon $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$2, then

$$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$3

is a regular subdivision of $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$4 (Cape et al., 19 Aug 2025). This makes it possible to concatenate local building blocks in exact-realizability constructions.

7. Variants, subtleties, and adjacent notions

Several subtleties recur in the subject. First, the tropical crossing number is sensitive to edge lengths. The genus-two theta graph shows that the invariant is neither lower nor upper semicontinuous in the edge lengths: the graph is planar generically, but the equilateral specialization has crossing number $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$5 (Cartwright et al., 2014). This dependence on metric data is precisely what motivates the finite-graph invariant $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$6, which minimizes over all possible metrics (Cape et al., 19 Aug 2025).

Second, tropical planarity is stricter than ordinary graph planarity. A graph may be planar in the classical sense yet still require tropical nodes for generic metrics, as seen in chains of loops with bridges (Cartwright et al., 2014). Conversely, in the isometric balanced embedding framework, every metric graph can be realized with exactly its classical crossing number (Campo et al., 2016). This suggests that the choice of ambient category—nodal immersions versus balanced embeddings after tropical modification—materially affects how crossing complexity is measured.

Third, the literature uses adjacent terminology that should not be conflated. The invariant called tri-plane crossing number for knotted surfaces in $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$7 is defined from bridge trisections and tri-plane diagrams by

$$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$8

and concerns 2-knots rather than tropical curves (Gong et al., 2 Jun 2026). This is a different diagrammatic crossing invariant despite the formal resemblance in name.

Within tropical geometry proper, the invariant connects directly to algebraic geometry. One application constructs a rational map $$\left\lvert \det \begin{pmatrix} v_1 & u_1 \ v_2 & u_2 \end{pmatrix}\right\rvert.$$9 for a smooth proper non-Archimedean curve $\mathbb{R}^n$00 with skeleton $\mathbb{R}^n$01 such that the tropicalization is an isometry on the skeleton up to $\mathbb{R}^n$02 crossings (Campo et al., 2016). Another lifting result produces totally faithful tropicalizations for nodal algebraic curves from immersed tropical curves with rational edge lengths (Cartwright et al., 2014). These results show that tropical crossing number is not only a combinatorial invariant but also a control parameter for faithful or almost faithful passage between algebraic curves and their tropicalizations.

The present state of the subject therefore combines exact low-genus classifications, asymptotic upper and lower bounds, realizability theorems for all prescribed values, and computational methods for concrete graphs. A plausible implication is that future progress will continue to depend on integrating Newton polygon combinatorics, graph invariants such as gonality and treewidth, and explicit subdivision search.