---
title: "Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants"
authors: ["Colin Defant", "Nathan Williams"]
year: 2026
venue: "arXiv:math.CO"
arxiv: "2608.06225"
doi: null
source: "https://arxiv.org/abs/2608.06225"
topics: ["curiosity", "combinatorics", "knot-theory"]
seed_rank: 1484
seed_batch: "curiosity-2026-09-01"
reviewed: "2026-09-01"
pool: "maths-foundations"
relevance_score: 10
lineage: coxeter-catalan
cites:
  - title: "Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants"
    url: "https://arxiv.org/abs/2608.06225"
    year: 2026
    arxiv: "2608.06225"
    doi: null
see:
  - "1124-a-categorification-of-the-jones-polynomial"
---

# Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants

## One-sentence takeaway

Run categorification backwards: the \((a,z{=}0)\)-HOMFLYPT polynomial is a reflection-subword count in the Hecke image of a pure braid, and full-twist reciprocity swaps noncrossing partitions with clusters.

## Why it's lovely

Why you might love this: Khovanov (1124) went up; Defant–Williams go *down*. Dual Garside rigidity survives as the leading \(q=1\) term of noncrossing Hecke coefficients, so combinatorics recomputes a knot polynomial and, for free, EL-shells the noncrossing lattice (full twist) while its inverse computes the homotopy type. Positive powers → noncrossing partitions; negative powers → clusters; crystallographic conjugacy yields the first reflection-subword models for rational Catalan objects.

## Key ideas

- Rewrite pure braids in dual-square generators \(\mathbb{T}\); expand in the Hecke algebra; leading noncrossing coefficients = signed reduced \(T\)-subwords (Theorem 4.6).
- Combinatorial reciprocity via \([\mathcal{T}_e]\mathrm{H}(\Delta^2\beta)=(-1)^{\mathrm{wr}}q^{\cdots}[\mathcal{T}_e]\mathrm{H}(\beta^{-1})\).
- Full twist \(\Delta^2=\prod \mathbbm{t}\) over \(c\)-sorting inversions → EL-shelling of \(\mathrm{NC}(W,c^{-1})\).
- Rational Catalan: conjugate \(c^{\pm p}\) back to \(c\) in Weyl type; produce (sometimes signed) subword models.

## Caveats

Rational models are type-by-type in crystallographic type; \(E_7/E_8\) still signed; noncrystallographic conjugacy fails. Only the \((a,z=0)\) HOMFLYPT slice, not the full two-variable polynomial. 65 pages of Coxeter–Catalan machinery — not a one-line trick.

## Implications for Broadside

Decategorification as a research move: when the library is drowning in higher categories, ask what combinatorial shadow still computes. Bridges knot/Hecke shelf to the Catalan objects already orbiting aperiodic/tiling cards.

## Links

- arXiv abs: https://arxiv.org/abs/2608.06225
- PDF: https://arxiv.org/pdf/2608.06225.pdf