7. Invariants of an Apparent Contour

By invariant, the authors of [33] mean a locally constant function on the set of stable mappings; see Sect. 7.3. See also [23, 32] and the references therein. The invariants considered in [33] turn out to be also invariants under diffeomorphic equivalence Definition 2.​4.​2).

With kind permission from Elsevier, in this section and in Sects. 7.2 and 7.3 we illustrate the results and report some of the figures from the quoted paper [9].

\(\mathfrak{B}(\text{appcon}(\varphi ))\) is automatically computed in the appcontour program (Chap. 10).

Compare also with Sect. 2.​5.​3.

An informal way to realize that \(\mathfrak{B}(\text{appcon}(\varphi ))\) is independent of the Morse description is the following. Suppose we are given two Morse descriptions of \(\text{appcon}(\varphi )\). Possibly composing with two elements of \(\text{Diff}_{\mathrm{c}}(\mathbb{R}^{2})\), we can suppose that the Morse lines of both the two Morse descriptions are horizontal and straight. The original apparent contour \(\text{appcon}(\varphi )\) is changed, under the action of these two diffeomorphisms, into two new apparent contours, say \(G\) and G′. Let us construct by hand an \(\mathbb{R}^{2}\)-ambient isotopy taking G into G′. One then classifies the events that appear in the path of diffeomorphisms taking G into G′, which are the following: local maxima or minima can be created or destroyed, and one checks that in both cases, (7.3) is unchanged. In addition, the number of crossings does not change, since a diffeomorphism of \(\mathbb{R}^{2}\) can only locally “rotate” and translate a crossing. When performing a local rotation, local maxima and minima are introduced, and one checks directly the invariance using definition (7.3) and Definition 7.1.1. Also the number of cusps does not change; moreover, cusps have been previously transformed into transversal marked points: applying a local rotation to an arc containing a marked point, the invariance follows from the definition, since the weight is independent of the orientation of the arc.

That is, the tangent line to \(\text{appcon}(\varphi )\) times \(\mathbb{R}\).

According to the vector product of the tangent vector to the path above and the tangent vector to the path below.

As we have already said in the Introduction, \(\mathit{BL}(\text{appcon}(\varphi ))\) is called a Bennequin-type invariant; see, e.g., [36]. In [33] it is proved that all local first order Vassiliev-type invariants of \(\text{appcon}(\varphi )\) are a combination of the number of cusps of \(\text{appcon}(\varphi )\), the number of crossings of \(\text{appcon}(\varphi )\), and of \(\mathit{BL}(\text{appcon}(\varphi ))\).

Or also the bifurcation set of \(\mathcal{C}^{\infty }(M, \mathbb{R}^{2})\) (see [42, 43] and [30]). Recall that, if a map \(\varphi\) belongs to \(\text{Unstable}(M, \mathbb{R}^{2})\), then every neighbourhood of \(\varphi\) contains maps not equivalent to \(\varphi\). We refer to the survey article [13] for more information.

Not surprisingly, such a classification is similar to the one in Chap. 6; this becomes reasonable, if one interprets \(\varphi\) as the first two coordinates of a local embedding of M in \(\mathbb{R}^{3}\). Notice carefully that M is, in this chapter, an abstract two-manifold, therefore there is no labelling on \(\text{appcon}(\varphi )\). In contrast, in Chap. 6 only labelled apparent contours are taken into account, and for this reason the number of possible cases is much larger.

See, e.g., [2‐4, 6, 7] and [22].

Indeed, it suffices to check that both functions are zero far from \(\text{appcon}(\varphi )\) and that both have the same jumps when crossing \(\text{appcon}(\varphi )\).

See also [17, Definition (1.19), p. 107] for related subjects.

Therefore, the resulting construction is not only invariant under diffeomorphic equivalence of apparent contours Definition 2.​4.​2), but also under \(\mathbb{R}^{3}\)-ambient isotopies.

See, e.g., [26], [35, Thm. 1] and [27].

By a vertex (respectively an edge) of \(\mathcal{P}\) we mean the homeomorphic image of a vertex (respectively an edge) of a closed planar polygon.

Recall that the Euler–Poincaré characteristic of a \(\mathit{CW}\) complex is \(\sum _{\text{d}}(-1)^{\text{d}}\#\{\text{d}-\text{dimensional cell}\}\); see, for instance, [40, p. 429], [11].

This graph contains the singular curve (see Remark 3.​2.​4). The corresponding partition has been considered, in more generality, for instance in [46].

Suppose that \(\mathcal{M}\) is a compact oriented connected three-manifold with boundary, and consider the double \(D(\mathcal{M})\) of \(\mathcal{M}\), obtained by identifying \(\partial \mathcal{M}\) with the boundary of a copy \(-\mathcal{M}\) of \(\mathcal{M}\), with opposite orientation. Then [37, p. 261] \(\chi (D(\mathcal{M})) = 0\). On the other hand, it is possible to show that \(\chi (D(\mathcal{M})) = 2\chi (\mathcal{M}) -\chi (\partial \mathcal{M})\).

Under \(\mathbb{R}^{3}\)-ambient isotopies with compact support.

Beware however that \(\mathbb{R}^{3}\setminus E\) and \(\mathbb{S}^{3}\setminus E\) are not, in general, homotopically equivalent; for example, the complement of a solid sphere in \(\mathbb{S}^{3}\) is contractible, whereas the complement in \(\mathbb{R}^{3}\) is not.

The Euler–Poincaré characteristic and the genus are related by \(\chi (\Sigma ) = 2 - 2g\).

The reason being that \(\Sigma \) separates \(E\) (the interior) from \(\mathbb{R}^{3}\setminus E\) (the exterior): the interior is below the ceiling, and consequently it cannot be above it at the same time.

There are exceptions, most notably the unknotting theorem [38, p. 103] asserts that “trivial” fundamental groups for \(\Sigma \) with the topology of a torus imply that the scene is ambient isotopic to the standard solid torus.

Proving that the trefoil knot cannot be deformed into its specular version, although apparently obvious, requires quite sophisticated techniques, beyond the scope of this book.

This is the most interesting choice in the case E is a knotted solid torus (tubular neighbourhood of a knot), or a union of knotted solid tori, tubular neighbourhood of a link. Indeed in this case the fundamental group of \(\mathbb{R}^{3}\setminus E\) is simply called the knot group (respectively link group), whereas the fundamental group of E simply reduces to \(\mathbb{Z}\), the infinite cyclic group, for each connected component. Of course in our broader context we can imagine situations, such as the sphere with a knotted tunnel of Fig. 10.20, where the interesting solid set is \(E\) itself.

Strictly speaking, \(x_{1},\ldots,x_{n}\) are free generators of the free group F of rank n; \(r_{1},\ldots,r_{m}\) are elements of F and G is the quotient G = F∕H where H is the smallest normal subgroup of F containing \(r_{1},\ldots,r_{m}\).

Although as a matter of fact the isomorphism problem is decidable if restricted to special classes of finitely presented groups. Among these, interestingly, we also find the fundamental groups of three-manifolds. A quite interesting post on this subject by Henry Wilton can be found at the web address http://​ldtopology.​wordpress.​com/​2010/​01/​26/​3-manifold-groups-are-known-right/​ (May 21,2014).

The related word problem (respectively conjugacy problem) of deciding whether two words define the same element (respectively conjugate elements) in a finitely presented group is also undecidable in general.

The rank r is actually equal to the Betti number b ₁ of the component C that we are considering. The other nonzero Betti numbers are b ₀, which is equal to 1, since we are restricting to a single connected component of \(\mathbb{R}^{3}\setminus E\), and b ₂: the number of “voids” (cavities) in C, equal to the number of connected components of the complement of C (which is also the number of connected components of \(\Sigma \) adjacent to C) decreased by one. The Euler–Poincaré characteristic of C is given by \(b_{0} - b_{1} + b_{2}\).

It can be defined for any matrix with entries in a principal ideal domain.

In [20] the emphasis is given to the ideal of L generated by the Alexander polynomial, indicated by \(\varepsilon _{1}\), see Sect. 7.7.

The mapping \(t \rightarrow 1/t\) corresponds to the automorphism of the ring \(\mathbb{Z}\) mapping the (multiplicative) generator t onto its inverse. This is the unique nontrivial automorphism of \(\mathbb{Z}\). Interestingly, it turns out that the Alexander polynomial of a knot is invariant under such a transformation, up to multiplication by a power of t; this is not the case for a generic finitely presented group with infinite cyclic commutator quotient. The symmetry of the coefficients of the Alexander polynomial is a nontrivial fact, consequence of the Poincaré Duality isomorphism. It is known that any Laurent polynomial having symmetric coefficients and that evaluates to ± 1 for t = 1 is the Alexander polynomial of some knot [25].

Recall that \(\mathbb{Z}X\) and \(\mathbb{Z}G\) are noncommutative rings.

This is always the case for G the first fundamental group of sets of the form \(\Sigma = \partial E\), E or \(\mathbb{R}^{3}\setminus E\). This follows using the Mayer–Vietoris exact sequence [24] on the two solid sets E, \(\mathbb{R}^{3}\setminus E\), and their common boundary \(\Sigma \). Indeed, let ρ > 0 and consider the open sets \(E_{\rho }^{+}:=\{ x \in \mathbb{R}^{3}: \text{dist}(x,E) <\rho \}\) and \(E_{\rho }^{-}:=\{ x \in \mathbb{R}^{3}: \text{dist}(x, \mathbb{R}^{3}\setminus E) >\rho \}\), so that \(\mathbb{R}^{3} = E_{\rho }^{+} \cup (\mathbb{R}^{3}\setminus E_{\rho }^{-})\). We have the short exact sequence \(0 = H_{2}(\mathbb{R}^{3}) \rightarrow H_{1}(E_{\rho }^{+} \cap (\mathbb{R}^{3}\setminus E_{\rho }^{-})) \rightarrow H_{1}(E_{\rho }^{+}) \oplus H_{1}(\mathbb{R}^{3}\setminus E_{\rho }^{-}) \rightarrow H_{1}(\mathbb{R}^{3}) = 0\). Hence \(H_{1}(E_{\rho }^{+} \cap (\mathbb{R}^{3}\setminus E_{\rho }^{-}))\) and \(H_{1}(E_{\rho }^{+}) \oplus H_{1}(\mathbb{R}^{3}\setminus E_{\rho }^{-})\) are isomorphic. Since H ₁(∂ E) is (for \(\rho > 0\) sufficiently small) isomorphic to \(H_{1}(E_{\rho }^{+} \cap (\mathbb{R}^{3}\setminus E_{\rho }^{-}))\) which is a direct product of copies of \(\mathbb{Z}\), it follows that also \(H_{1}(E_{\rho }^{+})\) and \(H_{1}(\mathbb{R}^{3}\setminus E_{\rho }^{-})\) are direct products of copies of \(\mathbb{Z}\), and the assertion follows.

Strictly speaking, the isomorphism between the corresponding group rings induced by the isomorphism between G∕G′ and \(\mathbb{Z}^{2}\).

Note that when the presentation is directly obtained from a knot/link diagram using the Wirtinger technique [38], then the relation of the generators with their projection is easily obtained: all generators associated with the same link component project to the same element, whereas generators associated with different link component (one per component) project onto a base of the quotient group.

More generally this is the case for the inside and the outside of \(\Sigma \) made of two connected components of genus 1, i.e., two toric surfaces.

The unknotting Theorem is valid more generally for links with m components, in which case the fundamental group is the free group of rank m.