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 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 $\mathbb{R}^3\hookrightarrow\mathbb{R}^6$. 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 order $1$ invariant of T. Ekholm.