Forbidden patterns in tropical plane curves

Abstract

Tropical curves in $${\mathbb {R}}^2$$ R 2 correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most five.

Tropically planar graphs

“…Since the original submission of this paper, [17] have presented several new criteria for ruling out troplanarity, which reduces the number from 28 to 8 . For example, they prove a result akin to Corollary 3.5 forbidding two loops in a row for graphs with g ≥ 6 , as suggested in an earlier draft of this paper; this alone rules out 18 of the pictured graphs of genus 6.…”

“…Fig 17. All non-troplanar genus 6 graphs which were not ruled by any known criterion prior to the work in…”

Characterization of tropical plane curves up to genus six

“…A unimodular triangulation of a genus 6 polytope (left), its dual graph (center), and the corresponding skeleton which has a double heavy cycle with two loops (right) that the skeleton as a trivalent graph is obtained from the dual tropical curve by deletion of degree one unbounded edges and smoothening the degree two edges, which are obtained after the said deletion and continuing this process till we obtain a trivalent graph. We refer the reader to [1,3,4] for further details about the duality and the graph theoretic operation to obtain the skeleton. In [1], computational techniques were employed to classify all troplanar graphs for lower genera g = 3, 4, and g = 5.…”

Smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$

“…It is an open question whether it can be realized on a tropical plane in R 3 or R 4 ( [6]). Let us consider the tropical plane V(f ) in R 3 , where f is the tropical polynomial f = x ⊕ y ⊕ z ⊕ 0. Explicitly, if we define x := (−1, 0, 0), y := (0, −1, 0), z := (0, 0, −1), w := (1, 1, 1), X := R ≥0 x, Y := R ≥0 y, Z := R ≥0 z, W := R ≥0 w, XY := R ≥0 x + R ≥0 y, and so on, then V(f ) is expressed as…”

“…There are three candidates for the edge E opposing to p: (0, 1)-(2, 2), (0, 2)-(3, 0) and (0, 3)-(2, 0). Note that the edge (0, 2)-(3,1) is not a candidate for E by Lemma 5E is (0, 1)-(2, 2), it is one of the edges of the triangle in K * * containing (1, 2 − ǫ). The point(1,2) is one of the vertices of this triangle, and hence by Lemma 5.12, the skeleton of C is not the lollipop graph in this case.…”