[Paper Review] Markov trace on Funar algebra

S. Yu. Orevkov|arXiv (Cornell University)|Jun 4, 2012

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper establishes a rigorous framework for constructing a Markov trace on the Funar algebra $K_n(\alpha,\beta)$, resolving a foundational gap in earlier work. By proving that the trace's well-definedness reduces to verifying a finite set of identities—using a novel algorithmic approach—it enables computation of the universal Markov trace on $K_\infty$, with explicit results for the $ (\alpha,\beta) = (0,0) $ case yielding $ K_\infty(0,0;\mathbb{Z}[u,v]) / \bar{R} \cong \mathbb{Z}[u,v]/I $, where $ I = (16, 4u^2 + 4v, 4v^2 + 4u, u^3 + v^3 + uv - 3) $.

ABSTRACT

Funar algebra $K_\infty=K_\infty(α,β;k)$ is the quotient of the group algebra over a ring $k$ of the braid group $B_\infty$ by two cubic relations: $σ_1^3-ασ_1^2+βσ_1-1=0$ and another one which involves $σ_1$ and $σ_2$. The universal Markov trace on $K_\infty$ is the quotient map $t$ of $K_\infty(α,β,k[u,v])$ to its quotient (as a $k[u,v]$-module) by trace relations $xy=yx$ and by Markov relations $σ_nx=ux$, $σ_n^{-1}x=vx$ for $x\in K_n$. It is easy to check that the quotient is of the form $k[u,v]/I$ for some ideal $I$ (i. e. that the trace $t$ is determined by $t(1)$). We give an algorithm to compute the ideal $I$ and we present the result of computations in some special cases. In the last section we discuss some properties of the resulting link invariant. This invariant for $β=0$, $k=GF(37)[α]$ detects the chirality of the knots $10_{48}$ and $10_{91}$ and it distinguish many other pairs of knots with equal HOMFLY polynomials.

Motivation & Objective

• To correct a critical gap in the proof of the existence of a Markov trace on the Funar algebra, as identified in prior works by Bellingeri, Funar, and others.
• To establish a finite, algorithmic criterion for verifying the well-definedness of the Markov trace on $K_n(\alpha,\beta)$, replacing the previously insufficient infinite verification process.
• To compute the universal Markov trace on $K_\infty(\alpha,\beta;\mathbb{Z}[\alpha,\beta,u,v])$ via a systematic reduction process, enabling specialization to known algebras like BMW and Hecke.
• To provide explicit computations of the trace invariants for specific parameter values, particularly $ (\alpha,\beta) = (0,0) $, and relate them to known link invariants such as the HOMFLY polynomial.
• To demonstrate that the resulting invariant $ P_{0,0;\mathbb{Z}} $ is a 2-torsion refinement of a specialization of the HOMFLY polynomial, with values differing by at most 8 for equivalent HOMFLY polynomials.

Proposed method

• Introduces a new algebraic structure, the $K_n(\alpha,\beta)$ algebra, defined as a quotient of the braid group algebra $kB_n$ by cubic relations on $\sigma_i$ and a complex braid relation involving $\sigma_i^{-1}$, replacing earlier formulations using $\sigma_i^2$.
• Defines a Markov trace on $K_\infty$ as a $k[u,v]$-linear map satisfying $ t(xy) = t(yx) $, $ t(xs_n) = u t(x) $, and $ t(xs_n^{-1}) = v t(x) $, with the trace values computed modulo the ideal $\bar{R}$ generated by these relations.
• Develops a theoretical algorithm (Theorem 2.4) to compute the universal trace by reducing the problem to verifying a finite number of identities in the algebra, using $K$-reductions of words in $\sigma_i^{\pm 1}$.
• Employs $K$-reductions: elementary reductions using the defining relations (1) and (2), and a finite set of 21 reduced monomials in $\sigma_1^{\pm 1}, \sigma_2^{\pm 1}$, to normalize elements in $kF_\infty^+$.
• Performs explicit computer-aided computations (Corollaries 2.5 and 2.6) to compute the trace invariants for $ (\alpha,\beta) = (0,0) $, yielding $ K_\infty(0,0;\mathbb{Z}[u,v]) / \bar{R} \cong \mathbb{Z}[u,v]/I $ with $ I = (16, 4u^2 + 4v, 4v^2 + 4u, u^3 + v^3 + uv - 3) $.
• Relates the resulting invariant to the HOMFLY polynomial via a homomorphism to a Hecke algebra $H_\infty$, showing that $P_{0,0;\mathbb{Z}/4\mathbb{Z}}$ is a specialization of the HOMFLY polynomial and that $P_{0,0;\mathbb{Z}}$ differs from it by at most 8 in value.

Experimental results

Research questions

• RQ1 Can the well-definedness of the Markov trace on the Funar algebra be rigorously established, given a previously identified gap in earlier proofs?
• RQ2 Is it possible to reduce the verification of the Markov trace's well-definedness to a finite, algorithmically computable set of identities rather than an infinite process?
• RQ3 What is the structure of the universal Markov trace on $K_\infty(\alpha,\beta;\mathbb{Z}[\alpha,\beta,u,v])$, and can it be computed explicitly for specific parameter values?
• RQ4 How does the resulting link invariant $P_{0,0;\mathbb{Z}}$ relate to the HOMFLY polynomial, and what is the nature of their difference?
• RQ5 Can the trace invariants be computed explicitly for specific knots and links, and what are their algebraic properties?

Key findings

• The paper establishes that the Markov trace on the Funar algebra is well-defined if and only if a finite, algorithmically verifiable set of identities holds, resolving a foundational gap in prior work.
• For the case $ (\alpha,\beta) = (0,0) $, the universal Markov trace on $K_\infty(0,0;\mathbb{Z}[u,v])$ is computed as $ \mathbb{Z}[u,v]/I $, where $ I = (16, 4u^2 + 4v, 4v^2 + 4u, u^3 + v^3 + uv - 3) $, with the computation verified up to 12 crossings.
• The invariant $ P_{0,0;\mathbb{Z}} $ is shown to be a 2-torsion refinement of a specialization of the HOMFLY polynomial: if two links have the same HOMFLY polynomial, their $ P_{0,0;\mathbb{Z}} $ values differ by at most 8.
• The trace $ P_{0,0;\mathbb{Z}/4\mathbb{Z}} $ is proven to be a specialization of the HOMFLY polynomial via a homomorphism to a Hecke algebra, with the explicit formula $ P_{0,0;\mathbb{Z}/4\mathbb{Z}}(u,v) = h(ij^2u^{1/2}v^{-1/2}, -i) $, where $ h $ is the normalized HOMFLY polynomial.
• The paper provides a complete set of trace invariants for all knots and links with up to 12 crossings, with the number of distinct invariants growing from 1 to 86 as crossing number increases from 0 to 12.
• The algorithmic framework allows for the universal computation of the Markov trace on the Funar algebra, with the potential to specialize to other algebras like BMW and Hecke, though full computation remains computationally prohibitive for generic parameters.