The p-Adic Temperley-Lieb Parameter: Cyclotomic Units, Markov Traces, and the p-Adic Jones Polynomial

In the paper's words

Establishes the Temperley-Lieb parameter as p-adic cyclotomic units and constructs the p-adic Jones polynomial.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. We construct a p-adic Markov trace on $\text{TL}_n(\delta)$ valued in $\mathbb{Q}_p(\zeta_{2p^k})$ and prove it satisfies all Markov axioms (spherical, Markov property), yielding a p-adic Jones polynomial $V_L^p(t) \in \mathbb{Z}_p[\zeta_{2p^k}]$ for any link $L$.

   from Abstract

2. The natural p-adic analog of a root of unity is a cyclotomic unit — an element of $\mathbb{Z}_p[\zeta_m]^\times$ — because: Cyclotomic units form the natural torsion subgroup in p-adic fields Their p-adic valuation encodes arithmetic information (Iwasawa's theorem) The structure of $\mathbb{Z}_p[\zeta_{p^k}]^\times$ is the p-adic analog of $S^1$ in $\mathbb{C}$

   from 1.3 Why Cyclotomic Units?

3. Theorem (Iwasawa, §5 of ): The group of cyclotomic units $C_K \subset \mathcal{O}_K^\times$ has finite index in $\mathcal{O}_K^\times$ for $K = \mathbb{Q}_p(\zeta_{p^k})$.

   from 2.2 Structure Theorem

4. Then the TL parameter: $$\delta = -A^2 - A^{-2}$$ belongs to $\mathbb{Z}_p[\zeta_{2p^k}]$ and satisfies: $\delta$ is a cyclotomic unit (belongs to $\mathbb{Z}_p[\zeta_{2p^k}]^\times$) $\text{tr}_{K/\mathbb{Q}_p}(\delta) = -2 \cos(2\pi/p^k) \cdot [K:\mathbb{Q}_p]$ $\delta \equiv 2 \pmod{(1 - \zeta_{2p^k})}$ (the reduction modulo the uniformizer recovers the classical limit) Proof.

   from 3.1 The Main Theorem

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The valuation $v_p(V_L)$ is a p-adic link invariant that the classical Jones polynomial cannot detect.

   from 5.3 p-Adic Valuation as a New Invariant

2. The hierarchical structure of p-adic valuations may enable more efficient approximations than the archimedean case. p-adic HOMFLYPT: The HOMFLYPT polynomial generalizes the Jones polynomial to two variables.

   from 9. Open Problems

Where to read

1. Abstract198 words · 1 min
2. 1. Introduction332 words · 1 min
3. 2. p-Adic Cyclotomic Units261 words · 1 min
4. 3. The TL Parameter as Cyclotomic Unit391 words · 2 min
5. 4. The p-Adic Markov Trace346 words · 2 min
6. 5. The p-Adic Jones Polynomial437 words · 2 min
7. 6. Connection to Iwasawa Theory271 words · 1 min
8. 7. Computational Verification254 words · 1 min
9. 8. Implications for p-Adic Anyons180 words · 1 min
10. 9. Open Problems140 words · 1 min
11. 10. Conclusion142 words · 1 min