This is really a knot theory problem: "Does the knot $7_{2}$ (which I think is the name of this knot) have polygonal number 7?" The answer is no: the crossover points (viewed from above) are spaced at some fixed ratio between the two towers they join, so if we number the heights of the towers cyclically $a_{0}, a_{1}, \ldots, a_{6}$ in the proper order, we get that in a valid configuration $p a_{n}+(1-p)a_{n+2}> p a_{n+3}+(1-p)a_{n+1}$, where we treat the indices modulo 7. Summing the seven resulting relations gives $\sum a_{i}> \sum a_{i}$, a contradiction. ("7" could be replaced by any odd number larger than 3.)