Lin–Wang type formula for the Haefliger invariant

Abstract

In this paper we study the Haefliger invariant for long embeddings R 4k−1 ֒→ R 6k in terms of the self-intersections of their projections to R 6k−1 , under the condition that the projection is a generic long immersion R 4k−1 R 6k−1 . We define the notion of "crossing changes" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in R 4k−1 . This formula is a higher-dimensional analogue to that of X.-S. Lin and Z. Wang for the order 2 invariant for classical knots. As a consequence, we show that the Haefliger invariant is of order two in a similar sense to Birman and Lin. We also give an alternative proof for the result of M. Murai and K. Ohba concerning "unknotting numbers" of embeddings R 3 ֒→ R 6 . Our formula enables us to define an invariant for generic long immersions R 4k−1 R 6k−1 which are liftable to embeddings R 4k−1 ֒→ R 6k . This invariant corresponds to V. Arnold's plane curve invariant in Lin-Wang theory, but in general our invariant does not coincide with order 1 invariant of T. Ekholm.Date: October 13, 2018. Key words and phrases. the space of embeddings, the Haefliger invariant, configuration space integral, finite type invariant, generic immersion.

Knotted families from graspers

“…Theorem 8.1 [23,Theorem B]. For any 𝑑 ⩾ 3 and 𝑛 ⩾ 1, the lowest non-vanishing homotopy group of the layer 𝖥 𝑛+1 (𝑀) in the Taylor tower for Emb 𝜕 (𝔻 1 , 𝑀) admits an isomorphism (32) We first make a few remarks on the objects that appear in the statement. Firstly, 𝖫𝗂𝖾 𝜋 1 𝑀 (𝑛) is the group of decorated Lie trees from Section 3.5.…”

“…Haefliger first used this link in the case 𝑑 = 6𝑘 and 𝑝 = 𝑞 = 4𝑘 − 1 in order to construct a generator of Emb(𝕊 4𝑘−1 , 𝕊 6𝑘 ) or Emb 𝜕 ( 𝔻 4𝑘−1 , 𝔻 6𝑘 ) , by an ambient connect sum analogous to the one for the classical trefoil. Sakai [32] defined finite-type invariants of such knots and showed that the Haefliger trefoil is detected by a type 2 invariant; an invariant can be defined by foliating by arcs and then taking 𝖾𝗏 2 ∶ 𝜋 4𝑘−2 Emb 𝜕 (𝔻 1 , 𝔻 2𝑘+2 ) → ℤ, see (7).…”

“…This forms Haefliger's Borromean rings [19]: $$$$\begin{equation*} L\colon \mathbb {S}^p\sqcup \mathbb {S}^q\sqcup \mathbb {S}^{2d-p-q-3}\hookrightarrow \mathbb {R}^d, \end{equation*}$$$$Haefliger first used this link in the case $$d=6k$$ and $$p=q=4k-1$$ in order to construct a generator of $$\operatorname{Emb}(\mathbb {S}^{4k-1},\mathbb {S}^{6k})$$ or $$\operatorname{Emb}_\partial \left(\mathbb {D}^{4k-1},\mathbb {D}^{6k}\right)$$, by an ambient connect sum analogous to the one for the classical trefoil. Sakai [32] defined finite‐type invariants of such knots and showed that the Haefliger trefoil is detected by a type 2 invariant; an invariant can be defined by foliating by arcs and then taking $$\mathsf {ev}_2\colon \pi _{4k-2}\operatorname{Emb}_\partial (\mathbb {D}^1,\mathbb {D}^{2k+2})\rightarrow \mathbb {Z}$$, see (7).…”