Braid group and leveling of a knot

Braid group and leveling of a knot

Sangbum Cho Department of Mathematics Education
Hanyang University, Seoul 133-791, Korea
scho@hanyang.ac.kr
Yuya Koda Department of Mathematics
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan
ykoda@hiroshima-u.ac.jp
 and  Arim Seo Department of Mathematics Education
Korea University, Seoul 136-701, Korea
arimseo@korea.ac.kr
July 1, 2019
Abstract.

Any knot in genus- -bridge position can be moved by isotopy to lie in a union of parallel tori tubed by tubes so that intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the -length. We show that the -length equals the level number. We then find braid descriptions for -positions of all -bridge knots providing upper bounds for their level numbers, and also show that the -pretzel knot has level number two.

2000 Mathematics Subject Classification:
Primary 57M25
The first-named author is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-201800000001768).
The second-named author is supported in part by JSPS KAKENHI Grant Numbers 15H03620, 17K05254, 17H06463, and JST CREST Grant Number JPMJCR17J4.

1. Introduction

A knot in the -sphere is said to be in genus- -bridge position, or simply in -position, with respect to a standard torus if splits the -sphere into two solid tori and , and each of and is a single arc that is properly embedded in and respectively, and is parallel into . A knot is called a -knot if it can be isotoped to be in -position with respect to some standard torus. In a collar of in , one may take parallel copies of the form and tube each of the two consecutive copies to obtain a surface of genus in . Then we say that a knot lies in -level position with respect to if and meets each of the tubes in two arcs connecting the two ends of the tube. Any knot in -position with respect to can be moved into a knot in -level position with respect to for some in a natural way. We call it a leveling of the -position. The minimum such over all the possible level positions with respect to is an invariant of the -position, called the level number of the position. In particular, a knot has a -position of level number one if and only if it is a torus knot.

The level number of a -position is equal to a Hempel-type complexity called the arc distance of the position. Given a -position of with respect to as above, the arc complex is the simplicial complex whose vertices are the isotopy classes of simple arcs in connecting the two points . Then the arc distance of the -position for is defined to be the minimum simplicial distance between the collection of vertices represented by arcs in from that are parallel to in and the analogous collection for . In fact, it was shown in [3] that the level number equals the arc distance in more general setting, taking a genus- Heegaard surface and the -position of with respect to rather than the torus and the -position with respect to .

On the other hand, any -position of a knot can be described algebraically using the braid group on two points in the torus. In fact, each of the -positions of a knot corresponds to a collection of the words of the “reduced” braid group, that is the quotient of the braid group by its center. Such a description of -positions was introduced and used in [4] to compute the slope invariants of -tunnels of knots. Using the words in the collection corresponding to a -position, we define a new numerical invariant, called the -length of the position. The main goal of this work is to show that the level number of a -position equals the -length of the position (Theorem 5.2).

In Section 2, we introduce the level position of a -knot and leveling of a -position. In Sections 3 and 4, the reduced braid group and the -length are defined, and then our main theorem is proved in Section 5. In Section 6, we carry out some explicit computations for -positions of -bridge knots. Specifically, we find braid word descriptions for all the -positions of each -bridge knot, and provide upper bounds for their level numbers. We also give a conjectural description of the -bridge knots with level number . In the final section, we show that each of the -positions of -pretzel knot has level number two and obtain their braid descriptions.

Throughout the paper, we denote by and a regular neighborhood of and the closure of respectively, where is a subspace of a polyhedral space. The ambient spaces will be always clear from the context. Finally, the authors are deeply grateful to Darryl McCullough for his valuable advice and comments.

2. Leveling and level number

Let be a standard torus in the -sphere which splits the -sphere into two solid tori and . We regard a collar of in as the product so that . Choose the numbers with , and denote the level torus in the collar by . We construct a closed orientable surface of genus in the collar as follows.

First choose disks in so that each is disjoint from , and denote by the tube for . Then, from the union , remove the interiors of and for to get a closed surface of genus . We call such a surface is a surface determined by . The subsurfaces and are -holed tori while for -holed tori. (See Figure 1.)

Figure 1. The genus- surface .

For an integer , a knot in the -sphere is said to lie in -level position with respect to if there exists a surface determined by such that and is a pair of arcs connecting the two boundary circles of the tube for . A knot lies in -level position if lies in a standard torus in the -sphere. By definition, when a knot lies in -level position with respect to for , then is a single arc properly embedded in if . If , then is a pair of disjoint arcs properly embedded in such that each arc connects the two boundary circles of .

Figure 2. Examples of leveling. The second one is a 2-level position of the figure-eight knot.

When a knot lies in -level position with respect to , can be put into a -position with respect to by pushing the arc into the solid torus . It is not hard to see that the converse is also true. That is, given a knot in -position with respect to , one can isotope , keeping in and in at all times, so that lies in -level position with respect to for some . This is also a direct consequence of Theorem 3.2 in [3]. Such a level position is called a leveling of the -position. Now we define an integral invariant which measures the complexity of a -position of a knot.

Definition 2.1.

The level number of a -position of is the minimum number of level tori over all the levelings of the -position. The level number of a -knot is the minimum level number over all -positions of .

3. The braid group and braid descriptions of -positions

In this section, we briefly review a geometric interpretation of the braid group on two points in the torus, and braid descriptions of -positions, which were introduced in [4]. Let be a torus and let . Fix two points and in , and denote by , and the points and and the slice respectively for each . Consider a pair of disjoint arcs properly embedded in such that each endpoint of the arcs is one of , , and , and each of the arcs meets each slice transversely in a single point. There is an obvious multiplication operation on the collection of such pairs defined by “stacking” two pairs. Two such pairs are called equivalent if there is an isotopy of such that

  1. ,

  2. for ,

  3. for and , and

  4. sends one pair to the other pair.

The equivalence classes are called braids, and the above multiplication operation induces a group structure on them. We call the group of all braids under the induced multiplication the -braid group on the torus. A finite presentation of the -braid group on the torus is well-known, which can be found in [1], [10] or in [4]. We rewrite it as:

To describe the generators , and geometrically, first fix oriented meridian and longitude curves and in . (For convenience, we use the same symbols and for the generators and the curves.) Denote by the point , and fix a point disjoint from . Choose an arc in connecting and , and meeting only in . There are four isotopy classes of such arcs in , and we choose such that leaves in the direction of negative orientation of and leaves in the direction of positive orientation of . See Figure 3.

Figure 3.

Then the generator (respectively ) can be represented by a pair of arcs in of which one arc connects the vertices and after sliding around (respectively ) in the direction of the orientation on (respectively ), while the other one is vertical as in Figure 4. The element is represented by a pair of arcs which are half-twisted as in Figure 5 (a). Precisely, we choose a disk in such that is properly embedded in and meets only in . Then the element is represented by a pair of spanning arcs of the cylinder such that the arc connecting and over-crosses the other one once. The relations can be verified directly.


Figure 4. (a) a pair of arcs which represents . (b) a pair of arcs which represents .

Figure 5. (a) a pair of arcs which represents . (b) a pair of arcs which represents . (c) a pair of arcs which represents .

Now we weaken condition for the isotopy to

.

That is, we do not require that each is the identity on for . We call the new equivalence classes of the pairs of arcs under this condition the reduced braids, and the group of all reduced braids the reduced braid group and denote it by . The fundamental group of the torus can be considered as a subgroup of the -braid group on the torus naturally. That is, is generated by and , which are represented by pairs of arcs described in Figure 5 (b) and (c). One can verify that is central in the -braid group on the torus, and that the reduced braid group is the quotient of the -braid group by . Thus by adding two relations and to the above presentation, we obtain the following.

Proposition 3.1.

The reduced braid group has the presentation

.

Now suppose that the torus for the definition of the braid group is a standard torus in the -sphere which splits the -sphere into two solid tori and , and that the product is a collar of in with . The curves and in bound meridian disks of and respectively. Then any word in defines a knot together with a -position of with respect to . That is, the knot is obtained by attaching two arcs and to a pair of arcs representing and then by pushing the arc slightly into . Then the knot lies in -position with respect to .

The knot is well defined, indeed if two words and are equivalent in (i.e. represent the same braid) then the resulting knots and are -isotopic (that is, isotopic by an isotopy of the -sphere preserving at all times). The knot is said to lie in braid position (with respect to ), and the word is called a braid description of the -position for (after fixing the arc ). On the other hand, any knot that lies in -position with respect to is -isotopic to a knot for some word in . In fact, we will see that any knot in level position with respect to can be repositioned to a knot by isotopy keeping in at all times (Lemma 5.1).

By the construction, we observe that is -isotopic to each of , , and . In general, if is a word containing only powers of and , and of and , then is -isotopic to for any word (of course, and are possibly empty words).

Definition 3.2.

Let and be words in the reduced braid group . We say that is -equivalent to if is equivalent to in for some words containing only powers of and , and of and . We write when is -equivalent to .

The knots and are -isotopic if and only if and are -equivalent (with respect to the torus ).

4. Torus words in the reduced braid group

In this section, we introduce the words of some special types in the reduced braid group , called the -words and then define the -length, a numerical invariant for -positions.

Consider the subspace of as a -dimensional simplicial complex. That is, the vertices are the lattice points , and every edge is of length and is either vertical or horizontal. Let be the line segment in connecting the origin and the point where and are nonzero, relatively prime integers. There are many shortest paths in the complex from to . Among them, we will choose a special one, denoted by , which is “close” to the segment in some sense.

Let be the sequence of rectangles bounded by four edges of the complex such that each intersects in its interior as in Figure 6. We assume that and have the vertices and respectively. Note that the length of the sequence of rectangles for the line segment from to is (that is, ).


Figure 6.
  1. If , choose all vertices of that are separated by from the lower right vertex of for each .

  2. If and , then choose vertices of as in the case of for each , and choose the two upper vertices of .

  3. If and , then choose vertices of as in the case of for each , and choose the two upper vertices of .

Notice that some vertices may be chosen more than once. Given the vertex , we have the unique shortest path in from to passing through the vertices chosen in (1), (2) or (3). Denote by this shortest path. Note that the path depends only on the choice of . We give a direction on each horizontal edge of so that it is directed to the right if , and to the left if . Similarly, each vertical edge is directed upward if , and downward if (see Figure 7). We call the directed path from to . We also regard the segments from to and to themselves as directed paths and respectively.


Figure 7.
Definition 4.1.

Let and be nonzero, relatively prime integers. The torus word of type is a word in determined by the directed path as follows.

  • Each horizontal edge gives a letter if it is directed to the right and a letter if directed to the left. Similarly, each vertical edge gives a letter if it is directed upward and a letter if directed downward. Then the torus word of type is the word read off along the directed path from to .

Define the torus words of type , , and to be , , and respectively.

For example, the four directed paths , , and in Figure 7 determine torus words of types , , and as , , and respectively. By definition, any torus word does not contain , and no torus word contains both and (respectively both and ) simultaneously.

Now we will describe a pair of arcs which represents a torus word in the reduced braid group as follows. Recall our basic setup for the definition of the reduced braid group in Section 3. Let be a regular neighborhood of the torus . Fix oriented meridian and longitude curves and in such that is the point , and the point is disjoint from . Choose an arc in connecting and , and meeting only in such that leaves in the direction of negative orientation of and leaves in the direction of positive orientation of . See Figure 3. We also choose a disk in such that is properly embedded in and meets only in . Under this setup the generators of the reduced braid group were described geometrically in Section 3. As usual, for any subspace of , we denote by the parallel copy in .

Consider a standard covering of the torus in which the pre-images of and are vertical and horizontal lines and respectively. Then the pre-image of is the lattice , and the pre-image of is disjoint from . We will assume that the pre-image of the disk intersects each rectangle bounded by four edges of only in the lower right vertex. That is, we give the orientation of from the left to the right, and the orientation of from the downside to the upside.

Recall that is the line segment in connecting the origin and the point where and are relatively prime integers, and is the directed path in . The path is isotopic in to the segment fixing the two endpoints (or is equal to if is or ). Furthermore, we can place the disk in so that is isotopic to without crossing the pre-image of the disk except the two endpoints and . That is, the arc divides each rectangle into two portions. Then we say that the disk lies in good position (with respect to ) if lies in the intersection of the images of the portions having the lower right vertex of . (If , both two portions of the first rectangle or the last rectangle have the lower right vertex of the rectangle. In this case, we just choose the lower portion of the rectangle divided by .)

We extend the covering of to the covering of naturally. Considering the directed path in as an embedding , for , we define the arc in . Let be the image of in . Then the arc is properly embedded in and connects to . Furthermore, the arc together with the vertical arc forms a pair of arcs which represents the torus word of type .

On the other hand, considering as an embedding , for , the arc is isotopic to the arc without crossing the pre-image of except the two endpoints and . Let be the image of in . Then is an arc connecting the points to which is isotopic to in without crossing except and . Note that the arc projects into a -torus knot in which is the image of the line segment . We summarize the above observation as follows.

Lemma 4.2.

Under the setup with and its covering in the above, a word in is a torus word of type if and only if is represented by a pair of arcs in satisfying:

  • ,

  • projects into the -torus knot in that is the image of the line segment in joining and , and

  • lies in good position, that is, is isotopic to in without crossing except the endpoints and .

Definition 4.3.

A -word in the reduced braid group is the empty word or a word of the form

,

where, is a torus word and is an integer (possibly ).

Not all words are -words. For example a nonzero power of is not a -word. But by definition any word in is equivalent to some -word. Given a nonempty -word , define to be the minimum number for all expressions as above. For the empty word , define to be .

Definition 4.4.

Let be a word in . The -length of , denoted by , is the minimal number over all the -words which are -equivalent to . The -length of a -position is the -length over all braid description of the -position. The -length of a -knot is the -length over all -positions of .

Every torus word has -length , and we have . Also since

and is a torus word. Observe that and are torus words while and are not. But all of them have -length . For example,

and is a torus word.

5. Algebraic Description of Leveling

In this section, we will show that the level number is actually equal to the -length. We will use the same notations for the definitions of leveling and the braid description in the previous sections.

Lemma 5.1.

Let be a knot which lies in -position with respect to a standard torus in the -sphere. If the -position has a leveling of an -level position with respect to , then there is a braid description of the -position such that is at most . Conversely, if is a braid description of the -position with -length , then there is a leveling of the -position with level tori.

Proof.

To begin with, we recall the previous setup for the braid description of a -position. The torus splits the -sphere into two solid tori and , and we fix the oriented meridian and longitude curves and of , meeting in a single point and bounding disks in and respectively. We choose a point in disjoint from , and an arc in connecting and , and meeting only in such that leaves in the direction of negative orientation of and leaves in the direction of positive orientation of as in Figure 3. We also choose a disk in such that is properly embedded in and meets only in . In the standard covering of the torus , the pre-images of and are vertical and horizontal lines and respectively. We may assume that the pre-image in of the disk in intersects each rectangle bounded by four edges of only in the lower right vertex. We also consider the standard covering of . As usual, for any subspace of or , we denote by the parallel copy in or in .

Now suppose that a given -position for a knot with respect to admits a leveling of -level position with respect to . We will assume . (The case of is similar but simpler.) Then lies in a genus- surface described as follows. We first regard a collar of in as the product rather than for convenience, and let . There are disks in such that each is disjoint from . We denote by the tube for . Then the surface is obtained from the union by removing the interiors of and for . The intersection is a pair of spanning arcs connecting the two boundary circles of the tube for . By an isotopy, we may assume the disk in is identified with the disk defined in the previous paragraph, and further the two arcs are exactly vertical ones, and (see Figure 8 (a)).

Figure 8. The reposition of .

The union of the arc and the arc is isotopic to a -torus knot in for some relatively prime integers an . By an isotopy, we further assume that the disk lies in good position with respect to the line segment in joining the points and as in Lemma 4.2.

We will reposition the arc by an isotopy fixing the portion of outside as follows. We consider the arc as an embedding in , for , connecting to . Then the arc can be moved by isotopy of to the arc that is the union of the two arcs and , fixing the outside portion of (see Figure 8 (b)). Next, by an isotopy of the arc , we move its top endpoint to the point along so that the resulting arc, denoted by , projects into the -torus knot in which is the image of the line segment in . Finally, adding the arc to the unions of arcs and , we have a repositioning of inside , fixing the outside portion of (see Figure 8 (c)). Denoted by the arc , the pair of arcs represents the torus word, say , of type by Lemma 4.2. We denote by again the knot after the repositioning of .

The next step is to reposition the arcs of in a similar way, to obtain a torus word corresponding to the arcs. We denote by and the two arcs of where has the endpoints in and in while has in and in . After an isotopy, we may assume the two arcs are exactly vertical ones, and (see Figure 9 (a)).

The arc is an arc properly embedded in with endpoints and . We choose an arc, denoted by , properly embedded in with endpoints and . Then the union of , , and is isotopic to a -torus knot in for some relatively prime integers and . By an isotopy again, we assume that the disk also lies in good position with respect to the line segment in joining the points and as in Lemma 4.2.

Figure 9. The reposition of .

We regard the arc as an embedding in , for , connecting to . Then the arc can be moved by isotopy to the arc , fixing the remaining part of (see Figure 9 (b)). Next, we isotope a small neighborhood of so that the tube is identified with the tube , and

  • the arc is moved to the vertical arc , and

  • the arc is moved to the arc, denoted by , joining and , and it projects into the -torus knot in which is the image of the line segment in .

See Figure 9 (c). Denoted by the arc , the pair of arcs represents the torus word, say , of type , by Lemma 4.2 again. We denote by again the knot after the repositioning of .

We continue the process of the isotopy to obtain torus words consecutively for each . For the final step, let us consider the collar of in . The disk was identified with the disk , and hence the arc has endpoints and in . The union of the arc and the arc is isotopic to the -torus knot in for some relatively prime integers an . By an isotopy again, we assume that the disk also lies in good position with respect to the line segment in joining the points and as in Lemma 4.2. We reposition the arc to an arc by isotopy fixing the portion of outside as in the above. We consider the arc as an embedding in , for , connecting to . See Figure 10 (a).

Figure 10. The reposition of .

Then the arc can be moved by isotopy of to the arc that is the union of the two arcs and , fixing the outside portion of (see Figure 10 (b)). Next, by an isotopy of the arc , we move its bottom endpoint to the point along so that the resulting arc, denoted by , projects into the -torus knot in which is the image of the line segment in . Finally, adding the arc to the unions of arcs and , we have a repositioning of inside (see Figure 10 (c)). Denoted by the arc , the pair of arcs represents the torus word, say , of type , by Lemma 4.2. Finally, we obtain a braid description

,

of the -position, where is the torus word of type for each and the integer is determined by the half twists of the portion for each .

Conversely, let be a braid description of the -position with -length . Then is written as in the above. We may assume that the word is represented by a pair of arcs in a collar of in . Then the knot is the union of the two arcs in the pair with the arcs in and in . By the reverse process of the above argument, we obtain a surface in in which is positioned in -level position with respect to . ∎

The following is our main result, a direct consequence of Lemma 5.1.

Theorem 5.2.

The level number of a -position equals the -length of the position. The level number of a -knot equals the -length of the knot.

6. Braid descriptions of -bridge Knots

In this section, we will discuss the braid descriptions and level numbers of the -positions for -bridge knots. The braid descriptions for -bridge knots were also introduced in [4], which we will refine in detail. A knot in the -sphere is called a -bridge knot if there is a -sphere such that splits the -sphere into two -balls and , and each of consists of two disjoint properly embedded arc in parallel into . We call such a decomposition for a -bridge knot a -bridge position for . It is well-known that the -bridge position for any -bridge knot is unique up to equivalence (see [9]). The -positions of -bridge knots can be described by their tunnels as we will see in the following.

A tunnel for a knot in the -sphere is a simeple arc such that and the exterior of is the genus 2 handlebody. A knot which admits a tunnel is called a tunnel number one knot. Any -knot is tunnel number one. In fact, once we have a -position of with respect to a standard torus which splits the -sphere into two solid tori and , it is not hard to find tunnels in and in for . The tunnel is an arc in whose endpoints lie in the interior of the arc such that is a regular neighborhood of (after pushing the endpoints of slightly into the interior of ). The tunnel in is described in a similar way. See Figure 11 (a). Such tunnels and are called -tunnels for (with respect to the standard torus of the -position).


Figure 11.

Now let be a -bridge knot and let be a -sphere which splits the -sphere into two -balls and , and each of consists of two disjoint properly embedded arc in parallel into . We denoted by and the two arcs in and by and the two arcs in as in Figure 11 (b). In the figure, the arc in is an arc whose endpoints lie in the interiors of and such that a regular neighborhood of is (after pushing the endpoints of and slightly into the interior of ). The arc in is an arc whose endpoints lie in the interior of such that is a regular neighborhood of the union of and the sub-arc of having the same endpoints of . The other arcs , , and are described similarly. It is immediate that the six arcs , , , , and are all -tunnels for the -bridge knot . Furthermore, it was shown in [6] in that every tunnel of a -bridge knot is isotopic to one of the six tunnels.

On the other hand, there is a natural way to obtain a -position from a -bridge position of a -bridge knot. First, we choose an arc, say , among the four arcs of