Height, trunk and representativity of knots
In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity.
First, we give a conterexample for the conjecture which states that the height is additive under connected sum of knots. We also define the minimal height of a knot and give a potential example which has a gap between the height and the minimal height.
Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles.
Finally, we remark on the difference among Gabai’s thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.
Key words and phrases:knot, height, trunk, representativity, waist
2010 Mathematics Subject Classification:Primary 57M25; Secondary 57M27
We study a knot in the 3-sphere via a standard Morse function . We derive two geometrical invariants of a knot, one is “height” from the vertical direction of , and another is “trunk” from the horizontal direction.
1.1. Height of knots
It is often difficult to determine how geometrically defined knot invariants behave with respect to connected sum. Some classical invariants are known to be predictably well-behaved such as genus and bridge number . While others are only conjectured to be well-behaved such as crossing number and unknotting number. Still others have been shown to exhibit complicated behavior with respect to connected sum such as tunnel number  and width . In this paper we study the behavior of height of a knot with respect to connected sum and show that this invariant best fits in the third category by demonstrating that height is not additive with respect to connected sum, giving a counterexample to Conjecture 3.5 of .
Let be an ambient isotopy class of knot in and let be the standard height function. If is a smooth embedding of knot type , is morse and all critical points of have distinct critical values, then we will write . Though an abuse of notation, we will also let denoted the image of the embedding.
Then the bridge number of is denoted and is defined to be the number of maxima of . The bridge number of , , is the minimum of over all . Schubert showed that the bridge number of a connected sum always satieties the equality
Bridge number is closely related to the width of a knot which was originally defined by Gabai and used in the proof of the property R conjecture . To define width, we first need some additional structure. If is a regular value of , then is called a level sphere with width . If are all the critical values of , choose regular values such that . Then the width of is defined by . The width of , , is the minimum of over all . We say that is a thin position for if and write where denotes the set of all thin positions of .
Based in part on Schubert’s equality, it was widely conjectured that the width of a connected sum always satieties the equality
Rieck and Sedgwick made progress on this conjecture when they showed that the above equality always holds when and are mp-small knots . Additionally, Scharlemann and Schultens showed that . However, Scharlemann and Thompson proposed counterexamples to the equality in  and Blair and Tomova proved that an infinite class of the Scharlemann-Thompson examples were counterexamples . However, there are alternative definitions of width for which width is well-behaved with respect to connected sum . In general, the best known inequalities for are
with each of these individual inequalities known to be equalities for certain choices of and .
To define the height of a knot, we first need to introduce the notion of thick and thin level. A level sphere for is called thin if the highest critical point for below it is a maximum and the lowest critical point above it is a minimum. If the highest critical point for below is a minimum and the lowest critical point above it is a maximum, the level sphere is called thick. As the lowest critical point of is a minimum and the highest is a maximum, a thick level sphere can always be found. Note that some embeddings will have no thin spheres. When this occurs the unique thick sphere is called a bridge sphere and the embedding is said to be a bridge position for .
Given such that is a thin position, we define the height of , denoted to be the number of thick level spheres for . Similarly, the height of a knot type is defined as
Alternatively, we will define the min-height of a knot type to be
Clearly, for all knots .
The basic properties of height have been studied in . Here, we would like to point out the relation between the height and essential planar surfaces. Bachman proved in  that if a knot or link has thin levels when put in thin position then its exterior contains a collection of disjoint, non-parallel, planar, meridional, essential surfaces. Let denote the maximal number of disjoint, non-parallel, planar, meridional, essential surfaces in the exterior of . Then, we obtain the following inequality.
We are interested in understanding how height behaves with respect to connected sum. On the other hand, it holds that and this inequality can take an arbitrarily gap generally. Indeed let be a knot with for . Then and .
It was remarked in  that the height is additive with respect to connected sum for meridionally small knots (cf. [25, Theorem 1.8]), and conjectured that for non-trivial knots and , always holds. By a similar argument, it follows that min-height is also additive with respect to connected sum for meridionally small knots. Hence, it is natural to ask if min-height is always additive with respect to connected sum. Our first results provide counterexamples to each of these conjectures by defining an infinite class of knots such that for every and the following Theorem holds.
Let and let be any two-bridge knot, then
Since the height of any two-bridge knot is one, Theorem 1.3 gives a counterexample to Conjecture 3.5 of  that for all knots and , . By Theorem 1.3, the additivity of height does not hold with respect to connected sum of knots. At this stage, we expect the following.
For any two knots , it holds that
For a knot given in Theorem 1.3, we have . But it is natural to think that the gap between the min-height and the height can be taken arbitrarily large in general.
There exists a knot such that .
1.2. Representativity and trunk of knots
To measure the “density” of a graph embedded in a closed surface, Robertson and Vitray introduced the representativity in  as the minimal number of points of intersection between the graph and any essential closed curve on the closed surface. This concept was applied to a knot in the 3-sphere in  and extended to a spatial graph in the 3-sphere in . Let be a closed surface containing the knot . We define the representativity of a pair as
where denotes the set of all compressing disks for . Moreover, we define the representativity of a knot as
where denotes the set of all closed surfaces containing .
The representativity measures the “spatial density” of a knot. We summarize the known values of representativity of knots.
if and only if is the trivial knot ([26, Example 3.2]).
for composite knots ([26, Example 3.6])
for knots with essential -string tangle decompositions ([26, Example 3.6])
for 2-bridge knots ([26, Example 3.4])
for -torus knots ([26, Example 3.3])
For a -pretzel knot , if and only if or ().
for alternating knots ()
for inconsistent cable knots with index ()
([26, Theorem 1.2]).
, where denotes the distortion of ().
We remark that the inequality (10) was used to show the above (1), (4), (5), (6), (7). In this paper, we refine the inequality (10).
As in , we define the trunk of a knot as
It follows by the definition that .
The bridge number of knots behaves as expected under taking connected sums, that is, Schubert proved that (). On the other hand, it was naturally expected that ([25, Conjecture 1.7]). Davies and Zupan showed in  that this is true, namely, for two knots and ,
In several cases, the trunk turned out to be useful. For example, it was shown in  that , where denotes the multiplicity index of . It was also shown in  that a knot is embeddable into -tube if and only if .
The following theorem refines [26, Theorem 1.2].
For any knot , we have
In the following, we introduce a “local trunk” of a knot, that is, the trunk of a tangle which lies in the pair of the 3-sphere and a knot.
Let be a tangle, where is a 3-ball and is proper ambient isotopy class of properly embedded arcs in . Let be a standard Morse function with a single maximal point and .
If is a smooth embedding in the proper ambient isotopy class of , is morse and all critical points of in the interior of have distinct critical values, then we will write .
We define the trunk of a tangle as
Then we obtain the next theorem which is a local version of Theorem 1.6.
Let be a knot admitting an essential tangle decomposition . Then we have
Corollary 1.8 ([27, Theorem 1.5]).
For a large algebraic knot , .
Let be a large algebraic knot (i.e. algebraic knot with an essential Conway sphere). Then, admits an essential tangle decomposition , where is a union of two rational tangles. It is easy to see that . By Theorem 1.7, we obtain . ∎
2. Proof of theorems
2.1. Proof of Theorem 1.3
In this subsection we utilize the results in  to give a lower bound on the height and min-height of some satellite knots.
The following theorem is Corollary 5.4 in .
Suppose is the standard height function and is a handlebody for which horizontal circles of with respect to constitute a complete collection of meridian disk boundaries. Then there is a reimbedding so that
is a Heegaard splitting of .
The proof of the following theorem is a slight variation on the proof of Corollary 6.3 of .
Suppose is an embedding of knot-type in an unknotted solid torus in . Suppose is a knotted embedding of and is an embedding of knot-type . If , then and .
Let such that has thick levels with respect to . Let be the image of under an isotopy taking to . We can additionally assume is Morse with respect to after this isotopy. For every regular value of , is an unlink in . By standard Morse theory and since is a torus, there exists a regular value such that has a component that is an essential loop in . Moreover, since is a knotted solid torus, is a meridian curve for . By Theorem 2.1, there is a reimbedding of that preserves height and results in being unknotted. Moreover, after a suitable choice of , we can assume that , see  for details. Since is height preserving, and have the same number of thick levels and . Since is a thin position for and , then is a thin position for . Hence, .
Alternatively, let such that has thick levels with respect to . By the same argument as give above, we can find a height preserving reimbedding and such that and . Hence, . ∎
It is interesting to note that Theorem 2.2 does not hold if the hypothesis of is omitted. For example, if is a 2-bridge knot, it is an easy exercise to show that and . However, declaring and meets all the hypotheses of Theorem 2.2 except . Additionally, in Figure 2 we give an example of a thin position for a knot-type embedded in a knotted solid torus together with an embedding of knot type contained in the unknotted solid torus such that . The embedding depicted in Figure 2 is a thin position of by Lemma 6.0.6 of . Hence . Since , then the embedding in the figure illustrates that no bridge position for is a thin position. Hence, and moreover since by . Thus, and .
In , Blair and Tomova construct an infinite collection of ambient isotopy classes of knots from the schematic depicted in the left-hand side of Figure 3 by inserting suitable braids into the boxes shown. By Theorems 12.1 and 12.2 of , for all , and any thin position for has exactly three thick levels of width and exactly two thin levels of width . Hence, for all . Note that if we consider the height function to be increasing from the bottom to the top of Figure 3, then, for suitable choices of , the left-hand side of the figure depicts a thin position for any knot in .
Proof of Theorem 1.3.
By , and Figure 3 gives a thin position for . Figure 3 demonstrates that . By Corollary 6.4 of , and, therefore, . Hence, Figure 3 demonstrates a thin position or with three thick levels. In particular, and .
Proof of Theorem 1.6.
Firstly, if is the trivial knot, then we have and , and hence the inequality of Theorem 1.6 holds.
Next, we will show that for a non-trivial knot , a height function and a closed surface containing ,
By taking maximal of the left-hand side and minimal of the right-hand side, we have
By perturbing relative to , we may assume that any critical point of is not on and is also in a Morse position with respect to . Since the genus of is greater than , there exists a regular value for such that contains at least two essential loops in . Take two loops of which are essential in and innermost in . Let be mutually disjoint disks in which are bounded by respectively. Then we have
Without loss of generality, we may assume that
By cutting and pasting if necessary, we may assume that . Thus, is a compressing disk for and we have
Let be a knot admitting an essential tangle decomposition , where is a tangle decomposing sphere. In the following, we show
Proof of Theorem 1.7.
Let be a closed surface containing . Let be a standard Morse function for . It is suffice to show that
We remark that is non-trivial since admits an essential tangle decomposition. Hence the genus of is greater than and we may assume that . Since and are essential in the exterior of , we may assume that each loop of is essential in . If consists of a single essential loop, then both and are surfaces with strictly positive genus. Then, there exists a regular value for such that contains at least two essential loops in for . Otherwise, consists of at least two essential loops and contains at least two essential loops in for . Similarly to Proof of Theorem 1.6, we obtain a compressing disk for in such that
3.1. Several versions of thin position
In the previous part of this paper, we considered Gabai’s thin position (), that is, the set of all position minimizing the width for chosen regular values . Then we have already established the following.
There exists a candidate knot in Figure 1 such that .
There exist two knots and in Theorem 1.3 such that .
There exists a candidate knot in  such that cannot be obtained in .
Every thinnest level sphere for is incompressible in the complement of a knot .
There exists a knot in  such that has a compressible thin level sphere.
Next, let be the set of all Morse positions of which have minimal critical points among all Morse positions. We say that a knot belonging to is in a minimal critical position. Similarly, we can define the MCP-height and the min-MCP-height of as
Note that for any knot . Then we have the following.
There exists a knot in Figure 4 such that .
There exist two knots and in Figure 4 such that .
There exists a candidate knot in  such that cannot be obtained in .
There exists a knot in Figure 4 such that has a compressible thinnest level sphere.
There exists a knot in Figure 4 such that has a compressible thin level sphere.
Finally, we define the ordered thin position , that is, a Morse position of which minimizes the lexicographical order of monotonically non-increasing ordered set , where is the number of points of intersection between thick level spheres and .
For example, the embedding in Figure 4 has the complexity , but it can be reduced to and we obtain an embedding .
Similarly, we define the OTP-height of as
Then we have the following.
For any knot , .
can be obtained in , as the first term of the monotonically non-increasing ordered set .
Every thinnest level sphere for is incompressible in the complement of a knot (by a similar argument to ).
There exists a candidate embedding in  such that and has a compressible thin level sphere.
In Figure 5, we summarize a relation on several versions of thin position. For each region, we give an example of an embedding in the corresponding subset of Morse embeddings. Each of these examples is conjectural with the exception of the 2-bridge embedding and the embedding from Figure 4, which can easily be verified. Potential examples of embeddings , , and are referred from . We have a potential example from Figure 1.
3.2. Trunk-to-height ratio
We propose that the definition of a proportion of a knot, like as a person’s waist-to-height ratio, BMI and so on. We define the proportion of a knot as
measures the slimness of the knot, and it is normalized by dividing by . Thus we have for any knot , and if and only if any thin position of is a bridge position. It holds by  that if , then . We remark that and may not be obtained in as noted in the previous subsection.
Similarly, we can also define the MCP-proportion and OTP-proportion of as
We can also consider another version by replacing with the average trunk , that is, the average of the intersection number of all thick level spheres and .
We expect that high distance knots have high proportions. For example, if a knot has a bridge sphere of distance greater than or equal to , then every thin position is a bridge position by using results of  and , and hence .
For a knot given in Figure 1, we conjecture that there exists an embedding and that
We also conjecture that there exists an embedding and that
For a knot given in Theorem 1.3, by , there exists in Figure 3 such that and is not in since there exists an embedding of with two thick levels of width and two thick levels of width . Since it is shown in  that every has exactly three thick levels, each of width , we have
We conjecture that
By Theorem 1.6, we obtain the following.
For any knot ,
where denotes the maximal number of maximal points of .
3.3. Waist and representativity
Theorem 1.6 is compared with the inequality between the waist and trunk of knots. We define the waist of a knot as
where denotes the set of all closed surfaces in , and denotes the set of all compressing disks for in (). Then, we have for the trivial knot since any closed surface in is compressible, and by considering the peripheral torus , for non-trivial knots. It is known that for -braid knots (), alternating knots (), almost alternating knots (), Montesinos knots (), toroidally alternating knots (), algebraically alternating knots (), and that for inconsistent cable knots with index , where is a companion knot for ().
Theorem 3.4 ([25, Theorem 1.9]).
For any knot , we have
Theorem 1.6 and 3.4 bear a close resemblance to each other. We expected in [26, Problem 26] that for any knot . For example, any alternating knots satisfy this inequality since () and (). However, it does not hold for composite knots in general. The waist behaves as expected under taking connected sums, that is, ([25, Proposition 1.2]). On the other hand, we have whenever and are non-trivial. This shows that the representativity of knots behaves dissimilarly to other geometric knot invariants.
3.4. Representativity and non-orientable spanning surfaces
Aumann proved that any alternating knot bounds an essential non-orientable spanning surface (). Indeed, he showed that both checkerboard surfaces are essential. Recently, Kindred proved in  that for any non-trivial alternating knot , which confirmed Conjecture 4 in . From these results, we expect that if a knot bounds an essential non-orientable spanning surface, then . However, we have the next proposition.
For any integer , there exists a knot with which bounds an essential once punctured Klein bottle.
Let be a genus two Heegaard splitting of . Take a loop on as shown in Figure 7. Note that bounds a Möbius band properly embedded in which is formed by a non-separating disk and a band. Let be a loop obtained from a train track on as shown in Figure 7, where . By adding a band along to , we obtain a once punctured Klein bottle properly embedded in and a knot .
It is easy to see that is -seamed with respect to a complete set of essential disks in , that is, has been isotoped to intersect minimally and for each pair of pants obtained from by cutting along , and for each pair of two boundary components of , there exist at least arcs of intersection in that connect that pair of boundary components. The following lemma can be proved by an elementary cut and paste argument.
If is -seamed with respect to a complete set of meridian disks in and be an essential disk in , then .
By this lemma, for any compressing disk for in , intersects at least points.
Finally, to obtain a knot with , we re-embed in so that is boundary-irreducible. Then there exists no compressing disk for in , and we have . ∎
We remark that there exists a knot which does not bound an essential non-orientable spanning surface ().
We would like to thank to Koya Shimokawa for suggesting to consider the Scharlemann–Thompson type thin position.
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