In this paper, the authors advance an ongoing program of connecting the theory of finite type invariants of classical knots to the Taylor tower for knots given by the theory of embedding (or manifold) calculus of functors. It is known that the “algebraic” Taylor tower classifies finite type invariants [I. Volić, Compos. Math. 142, No. 1, 222–250 (2006; Zbl 1094.57017)], but only over the rationals. In addition, type \(n-1\) invariants in that setting only appear in stage \(2(n-1)\) of the tower. It has thus been a long-standing conjecture [R. Budney et al., Adv. Math. 191, No. 1, 78–113 (2005; Zbl 1078.57011)] that integral type \(n-1\) invariants are in fact classified by the \(n\)th stage of the Taylor tower. Furthermore, it has been conjectured that the Taylor tower gives abelian group-valued invariants that are compatible with the connected sum of knots.
The main results in this paper are that, indeed, there is a group structure on the stages of the Taylor tower that is compatible with connected sum and that the Taylor tower contains type \(n-1\) invariants in stage \(n\). To establish these results, the authors employ some standard tools that have proved to be fruitful in the setting, like cubical and cosimplicial diagrams (and associated spectral sequences), but also bring in novel approaches, most notable of which are the action of the operad of little intervals \(\mathcal C_1\) and Habiro’s clasper surgery.
In a little more detail, the authors look at the map \[ \text{AM}^{\text{fr}}_n\longrightarrow \text{AM}^{\text{fr}}_{n-1} \] between stages of the Taylor tower for classical long knots modulo immersions, \(\text{Emb}^{\text{fr}}(\mathbb R, \mathbb R^3)\). They show that this is a fibration of spaces with \(\mathcal C_1\) action. The canonical map \[ \text{Emb}^{\text{fr}}(\mathbb R, \mathbb R^3) \longrightarrow \text{AM}^{\text{fr}}_n \] is then shown to be surjective on components and to preserve the \(\mathcal C_1\) action (and is thus multiplicative). Lastly, using Habiro’s characterization of finite type invariants via clasper surgery, the authors show that the map \[ \pi_0\text{Emb}^{\text{fr}}(\mathbb R, \mathbb R^3) \longrightarrow \pi_0\text{AM}^{\text{fr}}_n \] is an invariant of finite type \(n-1\).
The paper does not go as far as showing that the Taylor tower for knots modulo immersions gives all finite type invariants over the integers, but it offers a roadmap for establishing this result. One of the main ingredients would be the collapse of the spectral sequence associated to the Taylor tower.