Abstract
The aim of this chapter is to illustrate some interesting invariants of apparent contours and labelled apparent contours. These invariants can be numbers, groups, polynomials; invariance here means that the they are insensitive to certain transformations, that will be specified case by case.
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Notes
- 1.
- 2.
- 3.
\(\mathfrak{B}(\text{appcon}(\varphi ))\) is automatically computed in the appcontour program (Chap. 10).
- 4.
Compare also with Sect. 2.5.3.
- 5.
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.
- 6.
That is, the tangent line to \(\text{appcon}(\varphi )\) times \(\mathbb{R}\).
- 7.
According to the vector product of the tangent vector to the path above and the tangent vector to the path below.
- 8.
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 \(\mathit{BL}(\text{appcon}(\varphi ))\).
- 9.
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.
- 10.
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.
- 11.
- 12.
Indeed, it suffices to check that both functions are zero far from \(\text{appcon}(\varphi )\) and that both have the same jumps when crossing \(\text{appcon}(\varphi )\).
- 13.
See also [17, Definition (1.19), p. 107] for related subjects.
- 14.
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.
- 15.
- 16.
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.
- 17.
- 18.
- 19.
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 \(\chi (D(\mathcal{M})) = 2\chi (\mathcal{M}) -\chi (\partial \mathcal{M})\).
- 20.
Under \(\mathbb{R}^{3}\)-ambient isotopies with compact support.
- 21.
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.
- 22.
The Euler–Poincaré characteristic and the genus are related by \(\chi (\Sigma ) = 2 - 2g\).
- 23.
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.
- 24.
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.
- 25.
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.
- 26.
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.
- 27.
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 \(r_{1},\ldots,r_{m}\).
- 28.
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).
- 29.
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.
- 30.
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 \(b_{0} - b_{1} + b_{2}\).
- 31.
It can be defined for any matrix with entries in a principal ideal domain.
- 32.
- 33.
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].
- 34.
Recall that \(\mathbb{Z}X\) and \(\mathbb{Z}G\) are noncommutative rings.
- 35.
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.
- 36.
Strictly speaking, the isomorphism between the corresponding group rings induced by the isomorphism between G∕G′ and \(\mathbb{Z}^{2}\).
- 37.
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.
- 38.
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.
- 39.
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.
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Invariants of an Apparent Contour. In: Shape Reconstruction from Apparent Contours. Computational Imaging and Vision, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45191-5_7
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