Explain the Zero Tension Condition of a Tropical Curve $\mathcal{H}(x)$

Question

Right now I am studying tropical mathematics and I just arrived at a statement that I'm having trouble understanding. Here's how its stated:

For a tropic polynomial in two variables. $p$, the tropical curve $\mathcal{H}(x)$ is a finite graph embedded in the plane $\mathbb{R}^2$. It has both bounded and unbounded edges, all of whose slopes are rational, and the graph satisfies a zero tension condition around each node as follows:

1. Before the next part I have one question: What is a node referring to here?

Consider any node $(x,y)$, of the graph, which we may as well take to be the origin, $(0,0)$. (Why can we take this to be the origin?) Then the edges adjacent to this node lie on lines with rational slopes. On each such ray emanating from the origin consider the smallest nonzero lattice vector (what is a lattice vector referring to here?). Zero tension at $(x,y)$ means the sum of these vectors is $0$.

2. What does it mean by saying we can take this to be the origin?
3. What is a lattice vector referring to here?

Any other insights are appreciated.

EDIT: There is an example afterwards of $p(x,y) = a \odot x \oplus b \odot y \oplus c)$ **note this is equivalent to: $p(x,y) = min(a+x, b+y, c)$

Answer 1

I am very far from a specialist. I think the "node" at here means the "intersection points" of the lines of the graph. Generally you can think of this as a projection of the 3D graph to the 2D plane base.

Now what is the appropriate way to think about it? The tropical polynomial is going to be a piece-wise linear function on the whole plane. The intersection points should be the breaking points of the piece-wise linear functions. It is obvious that the set of minimum values shift if we add a constant to all the equations defining the tropical polynomial. Thus we can assume the "node" to be the origin without losing generality.

I think David Speyer's point is that over the "break-lines" that define the intersection of two different linear functions should have rational coordinates. So if you assume the line has equation $y=\frac{m}{n}x$, then with appropriate $x$ you will get $(x,y)=(n,m)\in \mathbb{Z}^{2}$. The "zero-tension" condition simply means the sum of the first integer coordinate along the all edges passing a node should be zero. I suspect a standard induction argument might prove it.