Lin–Wang type formula for the Haefliger invariant

In this paper we study the Haefliger invariant for long embeddings $\mathbb{R}^{4k-1} \hookrightarrow \mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\mathbb{R}^{4k-1} \looparrowright \mathbb{R}^{6k-1}$. 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 $\mathbb{R}^{4k-1}$. 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 $2$ 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 $\mathbb{R}^3 \hookrightarrow \mathbb{R}^6$. Our formula enables us to define an invariant for generic long immersions $\mathbb{R}^{4k-1} \looparrowright \mathbb{R}^{6k-1}$ which are liftable to embeddings $\mathbb{R}^{4k-1} \hookrightarrow \mathbb{R}^{6k}$. This invariant corresponds to V. Arnold’s plane curve invariant in Lin–Wang theory, but in general our invariant does not coincide with the order $1$ invariant of T. Ekholm.

Keywords

space of embeddings, Haefliger invariant, configuration space integral, finite type invariant, generic immersion

2010 Mathematics Subject Classification

57Q45, 57R40, 57R42, 58D10, 81Q30

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Published 3 December 2015