AI 中文总结
该研究针对纽结不变量，提出去范畴化的组合技术，关联对偶辫群生成元等与纽结不变量，证明$(a,z=0)$-HOMFLYPT多项式可通过该技术计算，还得到非交叉分划等相关模型。
AI 中文摘要
纽结理论中的大量工作都围绕着对纽结不变量进行范畴化，从而强化这些不变量。我们采取相反的方法：去范畴化，更常被称为组合学。我们引入一种技术，将对偶辫群生成元、纯辫的赫克像以及反射群中的因子分解问题与纽结不变量联系起来。我们证明$(a,z=0)$-HOMFLYPT多项式可作为此类问题的解来计算。该技术的灵感来源于Coxeter-Catalan组合学。例如，我们利用赫克代数中全扭结的像给出了非交叉分划格的EL-可壳性的新证明；在一种令人惊讶的组合互反性中，其逆元可计算同伦型。类似地，将我们的构造应用于全扭结的正幂次时会自然产生非交叉分划，而应用于负幂次时则会产生簇复形。在晶体学类型中，我们利用所有Coxeter元的共轭性，给出了有理非交叉Catalan对象的首个反射子字模型。我们的互反性给出了两个模型：一个推广非交叉分划，另一个推广簇。应用相同方法可得到两个（有理）非交叉停车模型。
英文摘要
Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. We introduce a technique that relates the dual braid group generators, the Hecke images of pure braids, and factorization problems in reflection groups to knot invariants. We prove that the $(a,z{=}0)$-HOMFLYPT polynomial can be computed as a solution to such a problem. This technique was motivated by Coxeter--Catalan combinatorics. For example, we give a new proof of EL-shellability of the noncrossing partition lattice using the image of the full twist in the Hecke algebra; in a surprising sort of combinatorial reciprocity, its inverse computes the homotopy type. Similarly, noncrossing partitions naturally arise from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword models for rational noncrossing Catalan objects. Our reciprocity gives two models: one generalizing noncrossing partitions, and one generalizing clusters. Applying the same method produces two (rational) noncrossing parking models.
Comments65 pages, 6 figures, 2 tables
