Knot invariants from XC-structures on the Sweedler algebra are trivial

An XC-algebra is the minimum algebraic structure needed to define a framed, oriented knot invariant and generalises Lawrence’s invariant obtained from ribbon Hopf algebras. In this note, we show that the knot invariant produced by any XC-structure on the Sweedler algebra is completely determined by the framing of the knot. Furthermore, we also exhibit explicit families of XC-structures on the Sweedler algebra that do not have a ribbon Hopf-algebraic origin.

💡 Research Summary

The paper investigates the knot invariants that arise from XC‑structures on the Sweedler algebra, a four‑dimensional non‑semisimple complex algebra generated by elements s and w with relations s²=1, w²=0, and sw=−ws. An XC‑algebra is defined as a triple (A,R,κ) where A is a k‑algebra, R∈A⊗A and κ∈A are invertible, satisfying four axioms (XC0)–(XC3). These axioms encode the algebraic counterpart of the rotational Reidemeister moves used in Lawrence’s universal invariant construction.

The author first treats the commutative case. Proposition 2.1 shows that for any commutative algebra A, any choice of invertible R and a square‑root of unity κ automatically satisfies the XC‑axioms, and every XC‑structure on A is of this form. Theorem 2.3 then proves that for any commutative XC‑algebra, the universal invariant Z_A(K) of a long knot K depends only on the framing fr(K): Z_A(K)=ν^{fr(K)} where ν=∑β_i κ α_i. Consequently, the invariant is trivial for 0‑framed knots.

The main focus shifts to the Sweedler algebra S_W. The standard ribbon Hopf structure on S_W has a one‑parameter family of universal R‑matrices R_λ that are triangular (R^{-1}=R_{21}) and a balancing element κ=s. Lemma 3.1 shows that any triangular XC‑algebra yields a framing‑only invariant, reproducing the known triviality of the Reshetikhin–Turaev invariant for S_W.

The novel contribution is the systematic construction of many non‑triangular XC‑structures on S_W. Examples 3.2–3.5 present one‑, three‑, five‑, and six‑parameter families of R and κ that satisfy the XC‑axioms but for which R^{-1} is not a scalar multiple of R_{21}. Direct calculations verify invertibility, non‑triangularity, and that the element ν is not central, demonstrating that these structures cannot arise from any ribbon Hopf algebra on S_W nor from endomorphism algebras of representations of ribbon Hopf algebras.

The central result, Theorem 3.6, asserts that every XC‑structure on S_W produces a framed knot invariant that depends solely on the framing: Z_{S_W}(K)=ν^{fr(K)}. The proof proceeds in several steps: (1) Lemma 3.8 characterises invertibility of generic elements and tensors in S_W; (2) Lemma 3.9 identifies the Jacobson radical J=span{w,sw} with J²=0; (3) Proposition 3.10 shows that all commutators