A triple coproduct of curves and knots
Noboru Ito, Takeshi Komatsuzaki
TL;DR
The paper introduces a triple coproduct for knots on surfaces that coherently blends intersection theory with three-component smoothing, producing a stable invariant $(\nu\otimes\mathrm{id})\circ\Delta$ and positioning it as the canonical commutative analogue of Turaev's cobracket within a skein-theoretic framework. It proves invariance under stable moves, derives a smoothing-weight normalization in the symmetric setting, and connects the construction to the affine index polynomial. Through explicit examples, it demonstrates strong distinguishing power, including infinite families of non-equivalent knots. Conceptually, the work offers a classical-limit perspective on skein quantization, clarifying the role of smoothing choices and the relation to $\text{Sym}^3(V)$ and to Turaev's theory. Overall, it broadens invariants for curves and knots on surfaces by providing a robust, uniquely determined commutative structure with skein-theoretic origins.
Abstract
We introduce a triple coproduct for knots on surfaces, providing a commutative framework that decomposes a single-component diagram into three components (Section 2). This construction is motivated by the interplay between intersection theory and the affine index polynomial, and extends these ideas to a three-component setting (Section 5). Building on Turaev's cobracket theory, we define an integer-valued invariant under stable equivalence by combining the coproduct with an intersection-theoretic function (Theorem 1). Unlike classical cobrackets, which often collapse distinct local configurations, our approach preserves combinatorial traces of smoothing choices, enabling fine-grained detection of local crossing patterns (Definition 4). In the symmetric tensor setting, Reidemeister invariance uniquely determines the relations in the word space (Equations (4), (5)) and canonically fixes smoothing weights, revealing an intrinsic simplicity behind the algebraic framework (Corollary 1). This uniqueness result positions our construction as the canonical commutative analogue of Turaev's non-commutative cobracket and clarifies its interpretation as a classical limit of skein quantization, extending the theoretical scope beyond previously known invariants (Section 6). Examples demonstrate substantial distinguishing power, separating an infinite sequence of knots arising from distinct smoothing choices and broadening the reach of existing invariants (Proposition 2).