A mathematical preprint reports a way to determine, from a link diagram, where colored gl_N link homology can be nonzero. Its main theorem says the homology vanishes outside a diagram-dependent window set by the diagram’s writhe w(D), the number k of local maxima and the number n of crossings. The absolute grading offset from m(N−m)w(D) must not exceed m(N−m)k plus [min(m,N−m)]²n. The theorem is formal under its stated assumptions, but the window is not guaranteed to be tight for every diagram or chosen web position.
The result is therefore a bound on where nonzero grading can occur, rather than a claim that every point in the window is reached. The manuscript’s theta-web example shows why the chosen shy position matters: the same example produces different bounds in different positions.
How a diagram becomes a bound
The proof begins by using planar isotopy to put the diagram into a form called shy. In that position, the colored homology complex is a tensor product of Rickard complexes. Each summand categorifies the gl_N evaluation of a shy web, and the shift between quantum and homological gradings is m(N−m)w(D).
Resolving crossings creates no new local maxima, so every resolution web retains the diagram’s k maxima on arcs labeled m. The proof then applies an energy estimate for shy gl_N webs. When the labels l_i at local maxima satisfy 2l_i≤N, the maximum quantum degree of the web’s evaluation is at most an energy made from the terms l_i(N−l_i) and half the sum of the vertex products a(v)b(v).
A separate part of the argument controls the rotation term by assigning each parallel-resolution curve to a global maximum. A maximum with label l_i contributes no more than l_i(N−l_i), using the constraint on the coloring subset.
The proof reduces the remaining color range to 2m≤N by reversing all component orientations. With a common component color, the correction term disappears and a bigrading-preserving isomorphism identifies the m and N−m colorings. The crossings, local maxima and writhe remain unchanged in this step.
Crossings supply the correction term through the vertex part of the energy. For a crossing resolution with a=b=m and 0≤i≤m, its four vertices contribute (m−i)(m+i)=m²−i², which is no more than m². Adding that contribution over n crossings produces the quadratic crossing term in the theorem.
A lower bound for Legendrian links
The preprint applies the diagrammatic result to a Legendrian link K. Its corollary says that the minimum value of j−i for which m-colored homology is nonzero is at least m(N−m)tb(K) minus [min(m,N−m)]²c(K). Here tb(K) is the Legendrian Thurston–Bennequin number, while c(K) is the minimum crossing number among front projections in the Legendrian link type.
The formula ties the lowest nonzero homological grading to the Legendrian number while retaining an explicit penalty based on crossing complexity. That correction is part of the stated result, so the corollary is a diagram-sensitive lower bound rather than a universal sharp value.
Where the estimate can improve
The theta-web example gives a bound of 2N−3 in the right shy position but only 2N−1 in the middle position. The contrast demonstrates both dependence on the chosen position and the absence of universal sharpness.
The manuscript also says the crossing-number correction is not optimal in general. In the case N=2 and m=1, standard diagrams of the Hopf link, the right-handed trefoil and the (2,5)-torus knot attain n−2.
A possible refinement is proposed but not proved: under a stated maximal-y vertex condition, the estimate could subtract 2a(v₀)b(v₀). The manuscript says that condition always holds for applications when m=1, but not in general.
The strategy could in principle be extended to differently colored links and colored spatial webs. However, the simple reduction to 2m≤N fails in the different-color case unless the correction term is tracked or every color satisfies 2l≤N.
A mathematical preprint with a disclosed correction
The document is identified as arXiv:2608.25676v1 [math.GT], dated 26 August 2026. It is presented as a preprint rather than a completed journal publication.
The work was partially supported by the Simons Collaboration grant on New Structures in Low-Dimensional Topology and a Simons Dissertation Fellowship. The manuscript also discloses that generative AI tools helped find and fix an error in the original proof of Proposition 6 and were used to improve the writing.
Paper data and sources
Original title: A diagrammatic grading bound for colored $\mathfrak{gl}_N$ link homology
Authors: Hongjian Yang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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