A triple coproduct of curves and knots

TL;DR

This paper introduces a novel triple coproduct for knots on surfaces that extends intersection theory and Turaev's cobracket, resulting in a new invariant capable of distinguishing complex knot configurations.

Contribution

It defines a canonical, commutative triple coproduct invariant that preserves combinatorial information and extends classical invariants to a more refined algebraic framework.

Findings

Defines a triple coproduct decomposing knots into three components.

Constructs an integer-valued invariant under stable equivalence.

Demonstrates the invariant's ability to distinguish an infinite sequence of knots.

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).