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Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial

Anthony Christiana, Ben Clingenpeel, Huizheng Guo, Jinseok Oh, Jozef H. Przytycki, Xiao Wang, Hongdae Yun

TL;DR

This work analyzes the low-dimensional Yang–Baxter homology $H_n(R_{(m)})$ for operators $R_{(m)}$ that yield the HOMFLYPT polynomial, showing that computing $H_n(R_{(m)})$ is governed by $n+1$ initial conditions and delivering explicit formulas for $H_3$ and $H_4$. By introducing duality, order-preserving filtrations, and carefully structured decompositions of the chain complex, the authors reduce the problem to a finite set of final complexes $C_\bullet^{jf}$ and compute $H_3$ exactly as $H_3(R_{(m)}) \cong k^{\frac{m(8-3m+m^2)}{6}} \oplus \left( \frac{k}{1-y^2} \right)^{\frac{(m^2-1)(5m-6)}{6}} \oplus \left( \frac{k}{1-y^4} \right)^{m(m-1)}$. The paper further derives a rank framework for $C_n^{mf}$ via a combinatorial function $\tilde{S}(n,m,u)$ linked to Stirling numbers and outlines a program for extending these results to higher homology, exploring torsion, and potential links to Khovanov-type homologies. Overall, the work provides a robust computational strategy for Yang–Baxter homology in this setting and sets the stage for deeper structural connections in low-dimensional topology.

Abstract

In this paper, we analyze the homology of the Yang-Baxter Operators $R_{(m)}$ yielding the HOMFLYPT polynomial, reducing the computation of the $n$-th homology of $R_{(m)}$ for arbitrary $m$ to the computation of $n+1$ initial conditions. We then produce the explicit formulas for the third and fourth homology.

Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial

TL;DR

This work analyzes the low-dimensional Yang–Baxter homology for operators that yield the HOMFLYPT polynomial, showing that computing is governed by initial conditions and delivering explicit formulas for and . By introducing duality, order-preserving filtrations, and carefully structured decompositions of the chain complex, the authors reduce the problem to a finite set of final complexes and compute exactly as . The paper further derives a rank framework for via a combinatorial function linked to Stirling numbers and outlines a program for extending these results to higher homology, exploring torsion, and potential links to Khovanov-type homologies. Overall, the work provides a robust computational strategy for Yang–Baxter homology in this setting and sets the stage for deeper structural connections in low-dimensional topology.

Abstract

In this paper, we analyze the homology of the Yang-Baxter Operators yielding the HOMFLYPT polynomial, reducing the computation of the -th homology of for arbitrary to the computation of initial conditions. We then produce the explicit formulas for the third and fourth homology.

Paper Structure

This paper contains 18 sections, 13 theorems, 50 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background on Yang-Baxter homology
  3. Precubic modules and their homology
  4. Yang-Baxter operators
  5. Duality and ordering
  6. Duality and strictly order-preserving maps
  7. Weakly order-preserving maps are chain maps
  8. Filtrations and splittings
  9. Filtration by alphabet size
  10. Further reductions by filtrations
  11. Computation of $H_3$ and $H_4$
  12. Computation
  13. Remark on $\mathop{\mathrm{rank}}\nolimits(C_n^{(mf)})$
  14. Future work, speculations, and conjectures
  15. Künneth subchain complex of $(C_n^{m},\partial_n)$
  16. ...and 3 more sections

Key Result

Proposition 2.1

For $\tau$ as defined above,

Figures (5)

  • Figure 1.1: Face map $d_{i,n}$.
  • Figure 1.2: Computational tree for $d_4^{\ell}(a,b,c,d)$ where $a\leq b \leq c \leq d$ and its image under $\tau_4$
  • Figure 3.1: Illustration of the fact that $C_\bullet^4 \cong C_\bullet^{1,0} \oplus 3C_\bullet^{2,1} \oplus 3C_\bullet^{3,2} \oplus C_\bullet^{4,3}$, where e.g. $C_\bullet^{3,1}$ is represented by (3,1) in the tree. Note the paths ending at $(j, j - 1)$ go left $m - j$ times and right $j - 1$ times, so there are $\binom{m - 1}{j - 1}$ such paths, as claimed.
  • Figure 4.1: The row operation matrix $P_4$ from the diagonalization of $\partial_4$.
  • Figure 4.2: The column operation matrix $Q_4$ from the diagonalization of $\partial_4$.

Theorems & Definitions (50)

  • Conjecture 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Example 1.9
  • Proposition 2.1
  • ...and 40 more