Refinement of the energy bound at a maximal trivalent vertex

Prove that, for a shy $gl_N$ web $\Gamma$ with at least one trivalent vertex, if the two thin edges at a trivalent vertex $v_0$ of maximal $y$-coordinate lie above the thick edge, then $\max\deg_q\Gamma_N\le e(\Gamma)-2a(v_0)b(v_0)$, thereby potentially improving the crossing-number term in the grading bound for colored $\mathfrak{gl}_N$ link homology.

Background

The paper proves an energy bound for the quantum grading of the Murakami–Ohtsuki–Yamada evaluation of a shy glNgl_N web and uses it to bound the supporting gradings of colored glN\mathfrak{gl}_N link homology. The resulting link-homology estimate contains a crossing-number correction term that the authors note is not optimal in general; for several standard diagrams with N=2N=2 and m=1m=1, a smaller term already suffices.

The authors propose a possible refinement of the web-level energy estimate when the two thin edges at a highest trivalent vertex lie above the thick edge. The proposed improvement depends on combining the rotation contribution with the vertex-weight term, because the two families of curves may have opposite rotation numbers. The refinement is not established by the argument given in the paper, leaving the stated inequality unresolved.

References

The missing term could come from a refinement of Proposition~\ref{prop:energy bound} of the following shape: if $V(\Gamma)\not=\emptyset$ and the two thin edges at the trivalent vertex $v_0$ of maximal $y$-coordinate lie above the thick edge, then \max\deg_q{\Gamma}_N\le e(\Gamma)-2a(v_0)b(v_0). The heuristic is that the flow condition makes the colors on the two thin edges at $v_0$ disjoint, so that the curves running through them should be counted as a single family of size $a(v_0)+b(v_0)$, replacing $a(N-a)+b(N-b)$ by $(a+b)(N-a-b)$. We do not prove this here: the argument of Proposition~\ref{prop:energy bound} bounds the rotation term and the vertex weights separately, and that is not enough, since the two families of curves in question are assigned to two different local maxima and may well have opposite rotation numbers, in which case the claim above fails for the rotation term alone and can only be recovered from the term $\pi(\sigma(l(v)),\sigma(r(v)))$.

— A diagrammatic grading bound for colored $\mathfrak{gl}_N$ link homology  (2608.25676 - Yang, 26 Aug 2026) in Remark labeled rem:sharper