Complexity of unknotting of trivial -knots.
We construct families of trivial -knots in such that the maximal complexity of -knots in any isotopy connecting with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of .
Here we can either construct as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots , consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat -simplices in its triangulation.
These results contrast with the situation of classical knots in , where every unknot can be untied through knots of complexity that is only polynomially higher than the complexity of the initial knot.
1. Main result.
Let be a PL-unknot in with crossings on one of its plane projection. The results of [Dyn] imply that can be isotoped to the standard unknot through PL-unknots with at most crossings. (See also [L] for further results in this direction.) On the other hand it was proven in [NW] that for each and each computable function there exists a trivial knot triangulated into (flat) -simplices such that any isotopy between and the trivial unknot that passes through -knots must pass through a knot that cannot be triangulated into less than simplices.
Alternatively, one can consider smooth embeddings of into (or ) and measure the complexity of knots as their ropelength (also known as crumpledness - see [N]) that was defined as , where is the volume of , and denotes the injectivity radius of the normal exponential map for . In other words, is the supremum of all such that any two normals to of length do not intersect. Informally speaking, one can think of as the maximal radius of a nonself-intersecting tube centered at . For this measure of complexity it will still be true that if , then there exists a polynomial upper bound for the complexity of knots in an optimal isotopy connecting an unknot with the standard unknot, and for the worst case complexity of knots in the optimal isotopies grows faster than any computable function.
It is natural to conjecture that the results of [NW] for will also hold in the case . Here we will prove that the complexity of untying of a trivial -knot can grow faster than a tower of exponentials of any fixed height of the complexity of the unknot.
For each positive and arbitrarily large there exists a trivial -knot with complexity in or such that any isotopy between this knot and the standard -sphere passes through -knots of complexity ( times). (In other words, one needs to increase the complexity more than any tower of exponentials of a fixed height of the initial complexity before the -knots can be untied.) Here “complexity” means either the number of flat -simplices in a triangulation of the original PL-knot and each of the intermediate -knots (and in this case intermediate unknots also must be PL), or, if the original knot and intermediate knots are smooth, the complexity of a knot can be defined as its ropelength, , where is the injectivity radius of the normal exponential map in the ambient .
In order to prove this theorem we first construct a sequence of finite presentations of . These finite presentations have certain algebraic properties that help to realize them as “visible” finite presentations of -knots of complexity comparable with the total length of the corresponding finite presentations. Moreover, these finite presentations have the following additional property: In each of them there exists a trivial element of length comparable with the total length of the presentation, , such that one needs to apply the relations at least ( times) in order to demontrate that this element is, indeed, trivial.
Then we prove that the finiteness/effective compactness of the set of trivial -knots of bounded complexity (modulo the group of transformations of generated by dilations and translations) implies that if all trivial -knots could be “untied” without a very large increase of complexity, then we would be able to contract any null-homotopic closed curve in the complement to the original -knot to a point through closed curves that are not much longer than the original one. But the algebraic property of the finite presentations of -knot groups explained in the previous paragraph implies that this is not the case.
2. Finite presentations of the trivial group.
Recall that the group that has the finite presentation with two generators and one relator is called the Baumslag-Solitar group. (Here and below we use the standard notation for .) Note that for each , and therefore the commutator . However, one needs to apply the relation times in order to demonstrate this fact in the most obvious way. In fact, it is well-known that there is no essentially shorter way to write as a product of the conjugates of the relator and its inverse. In other words, the Dehn function of the Baumslag-Solitar group is (at least) exponential. A proof of this fact can be found in [G] (and a sketch of another simpler proof using van Kampen diagrams can be found in [S]). The idea of the proof in the paper of Gersten is that the Baumslag-Solitar group is an HNN-extension and, therefore, the realization complex of the finite presentation will be aspherical. (Recall that the realization complex of a finite presentation has one -dimensional cell corresponding to each generator of the group and one -cell for each relator of the group.) So, its universal covering will be contractible, and, in particular, will have the trivial second homology group. Therefore, there will be a unique way to fill each null homologous -chain in the universal covering by a -chain. Each way to represent as the product of conjugates of the relator and its inverse corresponds to a filling of the lift of the loop corresponding to in the realization complex to its universal covering. Therefore, it must have the same number of -cells counted with multiplicities as the filling corresponding to the obvious presentation of as a product of conjugates of the relator and its inverse. It remains only to check that -cells in the universal covering that correspond to the obvious representaion of as the product of conjugates of the relator and its inverse do not cancel. Of course, the same proof implies that for each any representation of as a product of conjugates of the relator and its inverse has at least terms.
One can iterate the idea used in the construction of the Baumslag-Solitar groups and consider the following sequence of finite presentation of groups (see [B]). For each
This finite presentation has generators and relators of total length . One can prove that the Dehn function of grows as , where the height of the tower of exponentials is , using the approach of [Ger] (alternatively, one can use -bands) -see [B]. In particular, one can consider defined as . It is easy to see that ( times). Therefore, will be trivial. One needs to apply relations more than times to demonstrate that is trivial, when one proceeds in the obvious way. As above, one can use the asphericity of the representation -complex to conclude that the filling in its universal covering is unique on the -chain level. Below we will choose ”diagonalizing” this construction. Again, the same lower bound will hold for any non-zero power of with the same proof.
Now conider the following finite presentations , where are words considered in the previous paragraph. It is easy to see that is the finite presentation of , as in the corresponding Gromov group of total length . The following theorem is the key technical fact in our paper. It asserts that any way to demonstrate that, say, in would involve at least ( times) applications of the relations. Equivalently, each representation of as the product of conjugates of the relators and their inverses must involve at least this number of terms. We are stating and proving this fact using the language of van Kampen diagrams (cf. [LS]).
Each van Kampen diagram with the boundary in contains at least ( times) cells.
Consider a minimal van Kampen diagram with on the outer boundary. It must contain cells corresponding to the last relation. As there are no copies of on the boundary, these cells must form annuli (-annuli) (see Figure 1). Consider one of the innermost -annuli (that does not have any -cells inside). Its inner boundary must be a non-zero power of either or . The second option is impossible, as this would imply that the power of is trivial in , which is false. So, we have a non-zero power of on the innermost boundary. The part of the van Kampen diagram inside this innermost boundary is a van Kampen diagram in . But we already established that any such diagram must have size of at least ( times). ∎
3. Construction of a -knot.
Note that if one adds one more relator to the finite presentation , namely, , then one obtains a finite presentation of the trivial group. Denote the resulting finite presentation of the trivial group by . The finite presentation can be trasformed to the trivial finite presentation of the trivial group by performing elementary operations of the following types: 1) Replacing a relator by its inverse; 2) Replacing a relator by its product with another relator; and 3) Replacing a relator by or , where is a generator. Indeed, one can use operations involving only the relators and to replace these two relators by and , and then operations to transform each of the first relators to for an appropriate . (The trivial finite presentation here is the finite presentation, where the set of relators coincides with the set of generators). Note the , the considered trivial finite presentation and all intermediate finite presentations are balanced, that is the number of generators is equal to the number of relators.
For each balanced finite presentation of the trivial group we can construct a smooth -manifold in by starting from the connected sum of several copies of , where in each copy of corresponds to one of the generators, and then performing surgeries killing the relators. More precisely, we realize each relator by a simple closed curve , remove the tubular neighnorhood of , glue in a copy of so that its boundary is glued to the boundary of the removed tubular neighnorhood of , so that the boundary of is glued to a curve isotopic to , and smooth out the boundary. Alternatively, we could start from the -complex with one -cell, -cells corresponding to the generators and -cells corresponding to relators of (i.e. the realization complex of ), embed it into , take the boundary of an open neighborhood of and smooth-out the corners. The resulting smooth -manifold will have the fundamental group with the obvious finite presentation (in particular, it will be isomorphic to the trivial group), and the trivial second homology group. So, it will be a homotopy -sphere. Denote it by . But it is easy to see that will be diffeomorphic to . The reason is that if two balanced finite presentations and are related by one elementary operation of any of the three types introduced in the previous paragraph, the manifolds and are diffeomorphic via a diffeomorphism that can be described as a “handle slide”. Each elementary operation with finite presentation corresponds to an isotopy of a curve bounding a -disc forming an axis of a -handle. The isotopy of the boundary of a -disc can be extended to an isotopy of the -disc, the whole -handle and the whole -manifold. After finitely many operations we will end up with the -manifold constructed from a finite presentation with the same generator as and relators killing all the generators, which is diffeomorphic to the standard by means of an obvious diffeomorphism.
Thus, each handle slide as well as the whole sequence of handle slides used to constract diffeomorphisms between and can be regarded as an isotopy . This isotopy can then be extended to an obious isotopy between and the round sphere of radius one.
Before moving further we are going to give the following definition:
Let and be two positive valued functions defined on a closed unbounded subset of . We say that they have similar growth if there exist such that ( exponentiations both in the argument of and of the value of ) and ( exponentiations both of and ). here means . Increasing functions that do not have similar growth with (restricted to their domain) are called rapidly growing functions. An increasing function that is not rapidly growing is called reasonably growing.
Now note that our explanation of why is diffeomorphic to can be used to construct an explicit diffeomorphism such that its Lipschitz constant regarded as a function of is bounded by a reasonably growing function of .
Next consider the last relator, , and represent it by a simple curve in that we will also denote . It corresponds to a -handle in . The -disc filling forms a generator of this handle, which is diffeomorphic to . Consider the -sphere , where is a point inside . We claim that the fundamental group of the complement is isomorphic to , and, in fact, it is that this group has “apparent” finite presentation . Indeed, can be deformed to minus a tubular neighborhood of . Yet the deleted tubular neighborhood of decomposes into a -cell and a -cell. Therefore, if one attaches the deleted tubular neighborhood back then the “apparent” finite presentation of the fundamental group remains unchanged. Thus, is an “apparent” finite presentation of .
The meaning of ”apparent” finite presentation here is that each loop in can be homotoped to a bouquet of loops representing the generators of with an insignificant length increase. (Here and below a length increase is regarded as insignificant if it is measured by a reasonably growing function of .)
Consider a -knot in . Now consider a diffeomorphism between and that can be obtained as a sequence of handle slides corresponding to elementary operations transforming into . Take the composition of this diffeomorphism with a diffeomorphism between and the standard round sphere . Denote the resulting diffeomorphism between and the round by . Note that it is easy ensure that the Lipschitz constants of and its inverse were bounded by reasonably growing functions of . Indeed, it is sufficient to verify that this will hold for diffeomorphisms corresponding to the individual handle slides. Now each handle slide can be regarded as an isotopy extension that extends an isotopy of a simple curve bounding a -disc forming an axis of a -handle. One can discretize this isotopy of closed curve into small isotopes where the closed curves at the beginning and the end of the isotopy are normal variations of each other inside tubes of radius bounded by the injectivity radii of the normal exponential map of the curves. Further, one can ensure that the inverses of these injectivity radii of closed curves during the handle slide isotopies are uniformly bounded by a reasonably growing function of . Now it is obvious that Lipschitz constant of diffeomorphisms corresponding to the individual steps of the discretized isotopy are bounded by a reasonably growing function of . Now, it remains to check that the number of small steps in the discretization is also bounded by a reasonably growing function of . In other words, the isotopies do not need to be too long. In order to see this we can just analyze the isotopies corresponding to each of the elementary operations. (Alternatively, one can use an argument that provides an explicit upper bound for the number of points in a minimal -net in the space of Lipschitz curves of bounded length with injectivity radius of the normal exponential map bounded below by a positive parameter.)
The desired family of -knots are in the standard round . One can also perform a stereographic projection from a point on far from and obtain desired -knots in .
4. Filling functions.
Now we are going to use a concept from [Gro]: For each Riemannian manifold, or, more generally, length space if finite diameter we define its filling length function as follows. For each positive let denote the set of all contractible closed curves on that have length . For each let be the set of all homotopies contracting to a point. We consider elements of as one-parametric families of closed curves starting at and ending at a point. For each let denotes the maximal length of a closed curve in . Then define as the . Finally, we define filling of as the supremum of over all closed contractible curves . In order for this supremum to be finite it is helpful if all sufficiently short closed curves can be contracted to a point without length increase as it happens, for example, for Riemannian manifolds. Further, note that if is, in addition, simply connected, then can be majorized in terms of the value of this supremum on curves of length .
The choice of here is due to the well-known fact ([Gr]) that each closed curve in a Riemannian manifold (possibly with boundary) can be homotoped with almost no increase of length to a join of closed curves of length and contracted through closed curves of length length const. Indeed, one can choose any point , a finite set of points on such that and are close to each other, and to reduce contracting to contracting geodesic triangles . We see that if is simply-connected, then the length of will increase by at most for the described contracting homotopy, and then one can pass to the limit as .
In this paper we are going to apply this concept to spaces that are not simply connected, but have fundamental groups isomorphic to . These spaces will be complements to trivial -knots, and it is easy to see that the generators of can be represented by based loops of length that does not exceed twice the diameter of these spaces plus an arbitrarily small . Indeed,one can take a very small circle around the embedded and connect it with the base point by two minimizing geodesics travelled in the opposite directions. Denote the resulting curve by . Now proceed as in the simply connected case with the only difference that distances between points and on in the metric of are now chosen as rather than a very small . In this way we control the number of the triangles in terms of . Each of these triangles has length and is homotopic to a point or which is possibly iterated several times and maybe also travelled in the opposite direction. Once we homotope into a collection of integer iterates of (where the exponents must sum to zero), we will be able to cancel them and contract the resulting curve without increasing its length. Therefore, in order to prove the finiteness of it is sufficient to demonstrate the existence of the supremum of the maximal length of loops in an “optimal” homotopy contracting a loop of length to an integer power of . (Here the supremum is taken over all loops of length ; the word “optimal” means that we are taking the infimum over all such homotopies; the power of is uniquely determined by the initial curve and is locally constant). The existence of this supremum becomes evident when we combine the following two facts: First, note that the Ascoli-Arzela theorem implies the compactness of the set of closed curves in of bounded length parametrized proportionally to the arclength. Second, assume the existence of such that each closed loop of length can be contracted to a point without length increase. How let , be two loops such that for each . Then , will be homotopic to the same power of , and given a homotopy between and this power of , we can extend it to a homotopy for by merely adding a homotopy between and that does not increase length by much. This implies the second fact that the maximal length of loops in an “optimal” isotopy cannot significantly increase under (controllably) small perturbations of loops.
Now we plan to use these filling functions similarly to how it had been done in [N2]. The main idea is that they behave in a similar way to their algebraic counterparts that measure how difficult it is to see that all trivial words in a “visible” finite presentation of the fundamental group of are, indeed, trivial (or, more concretely, the maximum over all trivial words of a given length of the minimal area of a van Kampen diagram for the considered word). More specifically, we would like to consider complements to in , in the standard and to establish that 1) the values of for these complements are similarly growing functions of ; 2) for the complements of in is a not reasonably growing function of , as its growth is more or less the same as the growth of the area of van Kampen diagrams required to demonstrate that the groups of these -knots are trivial (see Theorem 2.1); and 3) If knots in the standard can be untied through -knots of not too high complexity, then the second of these two filling functions is reasonably growing. Taken together these three facts establish that the constructed -knots can be untied only through -knots of a very high complexity. One technical difficulty that arises here is the following: As the considered complements to -sphere are not compact, it is not clear that the values of for the considered complements are finite. More specifically, one can have contractible curves in, say, that include many very short arcs that go around in opposite directions. Also, the proof of the existence of given above used the existence of such that all closed curves of length are contractible (and even contractible by a length non-increasing homotopy). Therefore, we are going to remove not only the -knots but also their open tubular neighborhoods with radii given as the inverse values of a reasonably growing function of . More specifically, we proceed as follows.
In order to establish that cannot be untied through -knots of a not too high complexity we proceed by contradiction. We assume that in the standard can be isotoped to the standard unknot through -knots of complexity that is a reasonably growing function of . Consider the smooth case, when the complexity is defined as , where is the injectivity radius of the normal exponential map. (The proof in the simplicial case is quite similar.) First, we observe that that there is a reasonably growing function that majorizes not only for in the standard and in but also for all -knots in an isotopy connecting with an unknot in the round . (This fact follows from our assumption that can be untied through knots of not very high complexity.) Without any loss of generality we can normalize all metrics and assume that the areas of , , and all knots in an isotopy of to a round -sphere in a round -sphere are between one and , and also is a lower bound for the injectivity radii of all these -knots, where and are some reasonably growing functions of . For each of these -knots let denote its open tubular neighborhood of radius . We modify our idea and consider the complements to in and in a round rather than to and . In this way we obtain compact metric spaces and immediately see that their are finite. Moreover, the boundaries of these spaces are hypersurfaces (diffeomorphic to ) with principal curvatures bounded by a reasonably growing function of . The same will hold also for the complements of in the round , where denote the -knots in the considered isotopy between and a round -sphere. Now we are going to prove the following three lemmae:
The functions and are similarly growing functions of , where denotes the standard round sphere.
The assertion of the lemma immediately follows from the fact that Lipschitz constants of and its inverse are bounded by reasonably growing functions of . (We explained this fact at the end of the previous section.) This easily implies that and are also bi-Lipschitz homeomorphic with both Lipschitz constants bounded by reasonably growing functions. ∎
Now we are going to prove that:
is not reasonably growing.
The idea of our proof of Lemma 4.2 is that behaves essentially as the Dehn function(s) for the family of finite presentations . More precisely, assume that is reasonably growing. Then we are going to prove that there exist van Kampen diagrams for in with cells, where is a reasonably growing function. This will contradict Theorem 2.1.
Let us respresent by a sufficientlyly short loop in and contract it to a point via loops of length , where is a reasonably growing function of . Each of these intermediate loops can be first homotoped to a loop in the -skeleton of that can be regarded as the -dimensional Dehn complex corresponding to the finite presentation , and afterwards almost canonically represented as the product by of loops representing the generators of . “Almost canonically” means that the only ambiguities appear as the result of having different ways to represent the same loop by a small number of short words that correspond to relators of . The number of these small words is proportional to the length of and is, therefore, bounded by a reasonably growing function of . These amiguities will correspond to the discontinuities in our presentations of loops by words, and together will provide a representation of as the product of conjugates of words corresponding to these ambiguities. As the length of words is also bounded by a reasonably growing function of , so will be the number of cells in the corresponding van Kampen diagram. This completes the proof of Lemma 4.2. (Note that this argument is very similar to an analogous argument in the proof of Proposition 2.1 in our paper [LN].) ∎
Finally, we are going to prove that:
If the constructed -knots in can be untied through -knots of complexity bounded by a reasonably growing function, then is a reasonably growing function.
It is clear that Lemmae 4.1, 4.2 and 4.3 together immediately yield the contradiction that implies that the constructed family of -knots satisfies the conditions of our main theorem.
To prove Lemma 4.3 we assume that the constructed trivial knots can be untied through knots of complexity bounded by a reasonably growing function. As we are considering the smooth case, we can assume that the areas of the knots during the homotopy are between and , where is a reasonably growing function, and the injectivity radius of the normal exponential map during a contracting isotopy is bounded below by , where is also a reasonably growing function of .
Now our goal is to demonstrate that any closed curve of length in can be contracted with an increase of length bounded by a constant factor (say, ). If is not in -neighborhood of , then it is a convex metric ball of radius in and can be contracted within this ball without any length increase. Otherwise, it can be homotoped along outer normals to to the outer boundary of the -neighborhood of (or, equivalently, -neighborhood of ). Note that the length of each arc of under the projection increases less than by a factor of . It is easy to homotope to its projection that we denote through curves of length less than . (First, we continuously grow two copies of the normal between a point of and its projection on . Then we start moving one of the copies of the normal by moving its endpoints along and, correspondingly, . The intermediate curves consist of a constantly shrinking arc of , an expanding arc of and the two normals. Finally, this connecting normal returns to the original segment, and we cancel two copies of the original segment.) Now, one can contract inside a convex metric ball of radius in without a length increase.
This argument applies to the complement of -neighborhood of any -knot in such that its area and the injectivity radius of the normal exponential map satisfy the same bounds as the bounds for . Now we can adapt the argument from [N2] to prove that if two such -knots and with the injectivity radius of the normal exponential map greater than and volume between and are -close, then the values of for the complements of differ from each other by not more than a constant factor (say, ). The idea that in order to contract a contractible closed curve in, say, the complement to , one can transfer the curve to (the close in Gromov-Hausdorff metric) complement of without a significant length increase, contract the resulting curve there , discretize the contracting homotopy, transfer it back to the complement of and “fill” the discretized homotopy. In order for this program to work one need to be able to contract without a significant length increase “short” closed curves of length that does not exceed the distance between the complements to times an appropriate constant. Note that in order to see that is contractible in the complement of we can construct a contracting homotopy by similarly “transfering” a homotopy contractiung in the complement to . See [N2] for detailed descriptions of such transfers. This argument works, for example, if the distance between does not exceed .
Now note that the isotopy between the standard unknot and given unknot can be replaced by a sequence of “jumps” of “length” where the number of jumps is bounded by a reasonably growing function of . Here “length” means the Hausdorff distance between the considered knots. The number of these jumps is bounded by twice the number of pairwise disjoint metric balls of radius -net in the considered space of hypersurfaces in satisfying the same bounds for the volume and lower bounds () for the injectivity radius of the normal exponential map as for -knots . Indeed, we can replace any subsequence of “jumps” where the distance between the beginning and the end does not exceed by just one jump.
Now note that sizes of -nets in the space of hypersurfaces in with areas between and a reasonably growing function, and the injectivity radius of the normal exponential map bounded below by the inverse of a reasonably growing function is also bounded by a reasonably growing function. Such a bound will follow from any proof of the precompactness of the corresponding space of of hypersurfaces of bounded complexity and volume (cf. [N]). One possible idea is to represent these hypersurfaces as zero sets of appropriate -functions that vary in the same way along each normal segment to the hypersurface from and . It is easy to majorize -norms of these functions in terms of the available data, and to use a standard effective proof of the Ascoli-Arzela theorem (see [N] for the details of this argument).
Since the change of of the complements to the neighborhoods of -knots under such jumps does not exceed an explicit constant factor (cf. [N2], [LN]), we can start at the standard unknot , “jump” back to the given -knot and observe that for the complement ito its open neighbourhood of radius does not exceed , which is bounded by a reasonably growing function of .
Note that one can do a similar argument for -knots in instead of , but in order to have the desired compactness one needs to transfer all -knots during isotopies to a neighborhood of the origin (by appropriate translations). ∎
Again, one can easily adopt this proof for the PL -case.
Acknowledgements. This research has been partially supported by NSERC Accelerator and Discovery Grants of A. Nabutovsky.
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|Boris Lishak||Alexander Nabutovsky|
|Department of Mathematics||Department of Mathematics|
|University of Toronto||University of Toronto|
|40 St. George st.,||40 St. George st.,|
|Toronto, Ontario M5S2E4||Toronto, Ontario M5S 2E4|