Ropelength, crossing number and finite type
invariants of links
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of -component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates, and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates outperform the well known knot–genus estimate.
Given an –component link (we assume class embeddings) in –space
its ropelength is the ratio of length , which is a sum of lengths of individual components of , to reach or thickness: , i.e. the largest radius of the tube embedded as a normal neighborhood of . The ropelength within the isotopy class of is defined as
(in  it is shown that the infimum is achieved within and the minimizer is of class ). A related measure of complexity, called embedding thickness was introduced recently in , in the general context of embeddings’ complexity. For links, the embedding thickness of is given by a value of its reach assuming that is a subset of the unit ball in (note that any embedding can be scaled and translated to fit in ). Again, the embedding thickness of the isotopy class is given by
For a link , the volume of the embedded tube of radius is ,  and the tube is contained in the ball of radius , yielding
Bounds for the ropelength of knots, and in particular the lower bounds, have been studied by many researchers, we only list a small fraction of these works here [4, 5, 7, 12, 11, 10, 14, 29, 37, 39, 30, 38]. Many of the results are applicable directly to links, but the case of links is treated in more detail by Cantarella, Kusner and Sullivan  and in the earlier work of Diao, Ernst, and Janse Van Rensburg  concerning the estimates in terms of the pairwise linking number. In , the authors introduce a cone surface technique and show the following estimate, for a link (defined as in (1.1)) and a given component [7, Theorem 11]:
where is the maximal total linking number between and the other components of . A stronger estimate was obtained in  by combining the Freedman and He  asymptotic crossing number bound for energy of divergence free fields and the cone surface technique as follows
where is the asymptotic crossing number (c.f. ) and the second inequality is a consequence of the estimate , where is a minimal genus among surfaces embedded in , [17, p. 220] (in fact, Estimate (1.6) subsumes Estimate (1.5) since ). A relation between and the higher linking numbers of Milnor, [32, 33] is unknown and appears difficult. The following question, concerning the embedding thickness, is stated in [16, p. 1424]:
Let be an -component link which is Brunnian (i.e. almost trivial in the sense of Milnor ). Let be the maximum value among Milnor’s -invariants with distinct indices i.e. among . Is there a bound
for some constant , independent of the link ? Is there a bound on the crossing number in terms of ?
Recall that the Milnor –invariants of , are indexed by a subset of component indexes111where the last index plays a special role c.f. . . These are a well known link homotopy invariants (if all the indexes in are different in ) of –component links are often referred to simply as Milnor linking numbers or higher linking numbers, [32, 33]. The –invariants are defined as certain residue classes
where are coefficients of the Magnus expansion of the th longitude of in , and is a certain subset of lower order Milnor invariants, c.f. . Regarding as an element of , (or , whenever ) let us define
Our main result addresses Question A for general –component links (without the Brunnian assumption) as follows.
Let be an -component link and one of its top Milnor linking numbers, then
Naturally, Question A can be asked for knots and links and lower bounds in terms of finite type invariants in general. Such questions have been raised for instance in , where the Bott-Taubes integrals [3, 40] have been suggested as a tool for obtaining estimates.
Can we find estimates for ropelength of knots/links, in terms of their finite type invariants?
In the remaining part of this introduction let us sketch the basic idea behind our approach to Question B, which relies on the relation between the finite type invariants and the crossing number.
Note that since is scale invariant, it suffices to consider unit thickness knots, i.e. together with the unit radius tube neighborhood (i.e. ). In this setting, just equals the length of . From now on we assume unit thickness, unless stated otherwise. In , Buck and Simon gave the following estimates for , in terms of the crossing number of :
Clearly, the first estimate is better for knots with large crossing number, while the second one can be sharper for low crossing number knots (which manifests itself for instance in the case of the trefoil). Recall that is a minimal crossing number over all possible knot diagrams of within the isotopy class of . The estimates in (1.11) are a direct consequence of the ropelength bound for the average crossing number222i.e. an average of the crossing numbers of diagrams of over all projections of , see Equation (4.2). of (proven in [4, Corollary 2.1]) i.e.
In Section 4, we obtain an analog of (1.11) for –component links () in terms of the pairwise crossing number333see (4.10) and Corollary G, generally , as the individual components can be knotted. , as follows
On the other hand, there are well known estimates for in terms of finite type invariants of knots. For instance,
Lin and Wang  considered the second coefficient of the Conway polynomial (i.e. the first nontrivial type invariant of knots) and proved the first bound in (1.15). The second estimate of (1.15) can be found in Polyak–Viro’s work . Further, Willerton, in his thesis  obtained estimates for the “second”, after , finite type invariant of type , as
In the general setting, Bar-Natan  shows that if is a type invariant then . All these results rely on the arrow diagrammatic formulas for Vassiliev invariants developed in the work of Goussarov, Polyak and Viro .
Clearly, combining (1.15) and (1.16) with (1.11) or (1.14), immediately yields lower bounds for ropelength in terms of the Vassiliev invariant. One may take these considerations one step further and extend the above estimates to the case of the th coefficient of the Conway polynomial , with the help of arrow diagram formulas for , obtained recently in [8, 9]. In Section 3, we follow the Polyak–Viro’s argument of  to obtain
Given a knot , we have the following crossing number estimate
For a unit thickness knot ,
A somewhat different approach to ropelength estimates is due to Cantarella, Kusner and Sullivan. In , they introduce a cone surface technique, which combined with the asymptotic crossing number, , bound of Freedman and He,  gives
where the second bound follows from the knot genus estimate of : .
When comparing Estimate (1.19) and (1.18), in favor of Estimate (1.18), we may consider a family of pretzel knots: where is the number of signed crossings in the th tangle of the diagram, see Figure 1. Additionally, for a diagram to represent a knot one needs to assume either both and all are odd or one of the is even, .
Genera of selected subfamilies of pretzel knots are known, for instance [18, Theorem 13] implies
where , , are odd integers with the same sign (for the value of see a table in [18, p. 390]). Concluding, the lower bound in (1.18) can be made arbitrary large by letting , while the lower bound in (1.19) stays constant for any values of , , , under consideration. Yet another555out of a few such examples given in  example of a family of pretzel knots with constant genus one and arbitrarily large –coefficient is , , , where is the sign of (e.g. ). For any such , we have .
The paper is structured as follows: Section 3 is devoted to a review of arrow polynomials for finite type invariants, and Kravchenko-Polyak tree invariants in particular, it also contains the proof Theorem B. Section 4 contains information on the average overcrossing number for links and link ropelength estimates analogous to the ones obtained by Buck and Simon  (see Equation (1.12)). The proof of Theorem A is presented in Section 5, together with final comments and remarks.
Both authors acknowledge the generous support of NSF DMS 1043009 and DARPA YFA N66001-11-1-4132 during the years 2011-2015. The first author extends his thanks to Jason Cantarella, Jason Parsley and Clayton Shonkwiler for stimulating conversations about ropelength during the Entanglement and linking workshop in Pisa, 2012. The results of the current paper grew out of considerations in the second author’s doctoral thesis .
3. Arrow polynomials and finite type invariants
Recall from , the Gauss diagram of a knot is a way of representing signed overcrossings in a knot diagram, by arrows based on a circle (Wilson loop, ) with signs encoding the sign of the crossing; see Figure 2 showing the knot and its Gauss diagram. Given a Gauss diagram of a knot, the arrow diagrammatic formulas of [19, 35] are defined simply as a signed count of selected subdiagrams in . For instance the second coefficient of the Conway polynomial is given by the signed count of in , denoted as
where the sum is over all graph embeddings of into , and the sign is a product of signs of corresponding arrows in . For example in the Gauss diagram of knot on Figure 2, there are two possible embeddings of into the diagram. One matches the pair of arrows and another the pair , since all crossings are positive we obtain .
For other even coefficients of the Conway polynomial,  provides the following recipe for their arrow polynomials. Given , consider any chord diagram , on a single circle component with chords, such as , , . A chord diagram is said to be a -component diagram, if after parallel doubling of each chord according to , the resulting curve will have – components. For instance is a –component diagram and is a –component diagram. For the coefficients , only one component diagrams will be of interest and we turn a one-component chord diagram with a base point into an arrow diagram according to the following rule :
Starting from the base point we move along the diagram with doubled chords. During this journey we pass both copies of each chord in opposite directions. Choose an arrow on each chord which corresponds to the direction of the first passage of the copies of the chord (see Figure 3 for the illustration).
We call, the arrow diagram obtained according to this method, the ascending arrow diagram and denote by the sum of all based one-component ascending arrow diagrams with arrows. For example and is shown below (c.f. [9, p. 777]).
In , the authors show for , that the coefficient of the Conway polynomial of equals
Given a knot , we have the following crossing number estimate
Given and its Gauss diagram , let index arrows of (i.e. crossings of a diagram of used to obtain ). For diagram term in the sum , and embedding covers a certain element subset of crossings in we denote by . Denote by the set of all possible embeddings , and
Note that for for and for , thus for each we have an injective map
where . extends in an obvious way to the whole disjoint union , as , and remains injective. In turn, for every we have
Further, each arrow in indexed by , either agrees with the orientation of the Wilson loop (then we say it is a right arrow) or not (then it is a left arrow), thus , where is the subset of left arrows and the subset or right arrows with cardinalities and , . Since each arrow diagram has exactly right arrows and left arrows we must have
The left hand side is maximized for and thus we obtain
which proves the first inequality in (3.3). The second inequality is a simple consequence of Stirling’s approximation: . ∎
Next, we turn to arrow polynomials for Milnor linking numbers. In  Kravchenko and Polyak introduced tree invariants of string links and established their relation to Milnor linking numbers via the skein relation of Polyak . In the recent paper, the authors666consult  for a related result  showed that the arrow polynomials of Kravchenko and Polyak, applied to Gauss diagrams of closed based links, yield certain –invariants (as defined in (1.8)). For a purpose of the proof of Theorem A, it suffices to give a recursive definition, provided by the authors in , for the arrow polynomial of denoted by . Changing the convention, adopted for knots, we follow ,  and use vertical segments (strings) oriented downwards in place of circles (Wilson loops) as components.
The polynomial is obtained from by expanding each term of through stacking elementary tree diagrams and , shown on Figure 4. We begin with the initial tree , shown on Figure 4(right), and expand by stacking and on the strings of , this is shown on Figure 5, we avoid stacking on the first component (called the trunk, ). Thus is obtained as , where , , and . The sign of each term is , number of arrows pointing to the left, and denotes stacking onto the th string. Also, when obtaining terms of , during the stacking process, we must pay attention and eliminate isomorphic (duplicate) diagrams.
where , a Gauss diagram of an –component link , and the indeteminacy is defined in (1.8). For , we obtain the usual linking number
For and the arrow polynomials can be obtained following the stacking procedure as follows
Given a formula for all remaining –invariants with distinct indices can be obtained from the following permutation identity (for )
where is the arrow polynomial obtained from by permuting the strings according to .
One of the properties of –invariants is their cyclic symmetry, [33, Equation (21)], i.e. given a cyclic permutation , we have
4. Overcrossing number of links
We will denote by a regular diagram of a link , and by , the diagram obtained by the projection of onto the plane normal to a vector777unless otherwise stated we assume that is generic and thus a regular diagram . For a pair of components and in , define the overcrossing number in the diagram and the pairwise crossing number of components and in i.e.
In the following, we also use the average overcrossing number and average pairwise crossing number of components and in , defined as an average over all , , i.e.
Given a unit thickness link , and any –component sublink :
for , .
Consider the Gauss map of and :
If is a regular value of (which happens for the set of full measure on ) then
i.e. stands for number of times the –component of passes over the –component, in the projection of onto the plane in normal to . As a direct consequence of Federer’s coarea formula  (see e.g.  for a proof)
where is the normalized area form on the unit sphere in . Assuming the arc–length parametrization by and of the components we have and therefore:
where is often called illumination of from the point , . Following the approach of , and  we estimate . Denote by the ball at of radius , and the length of a portion of within the spherical shell: , where of and the radius . Note that, because the distance between and is at least , the unit thickness tube about is contained entirely in for big enough . Clearly, is nondecreasing. Since the volume of a unit thickness tube of length is , comparing the volumes we obtain
For a given ; , using the monotone rearrangement of , , we further obtain
and integrating with respect to the –parameter, we obtain
From the Gauss linking integral
thus we immediately recover the result of  (but with a specific constant):
Summing up over all possible pairs: , , and using the symmetry of the linking number we have
From Jensen’s Inequality , we know that and , therefore
Analogously, using the second estimate in (4.7) and Jensen’s Inequality, yields
Let be an -component link (), then
In terms of growth of the pairwise linking numbers , for a fixed , the above estimate performs better than the one in (1.5). One may also replace with the isotopy invariant888satisfying .
we call the pairwise crossing number of . This conclusion can be considered as an analog of the Buck and Simon estimate (1.11) for knots, and is stated in the following
Let be an -component link, and its pairwise crossing number, then
5. Proof of Theorem A
The following auxiliary lemma will be useful.
Given nonnegative numbers: , …, we have for :
It suffices to observe that for the ratio achieves its maximum for . ∎
Let be an -component link of unit thickness, and one of its top Milnor linking numbers, then
Let be a Gauss diagram of obtained from a regular link diagram . Consider, any term of the arrow polynomial: and index the arrows of by , in such a way that is the arrowhead and is the arrowtail, we have the following obvious estimate:
Let , since every term (a tree diagram) of is uniquely determined by its arrows indexed by string components, gives an upper bound for the number of terms in . Using Lemma H, with