Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres
We give a complete characterization of those essential simple loops on -bridge spheres of -bridge links which are null-homotopic in the link complements. By using this result, we describe all upper-meridian-pair-preserving epimorphisms between 2-bridge link groups.
2010 Mathematics Subject Classification:Primary 57M25, 20F06
The first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2009-0065798). The second author was supported by JSPS Grants-in-Aid 18340018 and 21654011, and was partially supported by JSPS Core-to-Core Program 18005.
For a knot or a link, , in , the fundamental group of the complement is called the knot group or the link group of , and is denoted by . For prime knots, the knot groups are complete invariants for the knot types (see ). Moreover we have a partial order on the set of prime knots, by setting if there is an epimorphism (see, for example, [32, Proposition 3.2]). Epimorphisms among link groups have received considerable attention and they have been studied in various places in the literature (see [1, 2, 5, 6, 10, 11, 15, 16, 17, 25, 27, 30, 32, 33, 34, 35, 36] and references therein).
In , a systematic construction of epimorphisms between 2-bridge link groups was given. The construction is based on a systematic construction of essential simple loops on -bridge spheres of -bridge links which are null-homotopic in the link complements. Thus the following question naturally arises (see [25, Question 9.1(2)]).
Let be a 2-bridge link, and let be a -times punctured sphere in determined by a -bridge sphere. Then which essential simple loops on are null-homotopic in ?
It should be noted that each -bridge link admits a unique -bridge sphere up to isotopy (see ), and hence the -times punctured sphere in the above problem is unique up to isotopy.
In this paper, we give a complete answer to the above question (Main Theorem 2.3). In fact, we show that those essential simple loops on constructed in [25, Corollary 4.7] are the only essential simple loops on which are null-homotopic in the -bridge link complement. This enables us to describe all epimorphisms between -bridge link groups which map the upper meridian pair of the source group to the upper meridian pair of the target group (Main Theorem 2.4). In fact, this theorem says that any such epimorphism is equivalent to that constructed in [25, Theorem 1.1].
To the authors’ knowledge, every known pair of -bridge knots with belongs to the list in [25, Theorem 1.1]. Kitano and Suzuki and their coworkers verified this for -bridge knots up to 11-crossings in [10, 15, 16]. Gonzaléz-Acũna and Ramírez  determined the -bridge knots whose knot groups have epimorphisms to the torus knot group, and their result implies that every such -bridge knot group is isomorphic to one constructed in [25, Theorem 1.1]. In their recent work , Boileau, Boyer, Reid and Wang proved Simon’s conjecture (see [14, Problem 1.12(D)]) for -bridge knot groups, namely they have shown that each -bridge knot group surjects onto only finitely many distinct knot groups. To be more precise, they have shown that if a -bridge knot group surjects onto a non-trivial knot group , then is a -bridge knot and the epimorphism is induced by a map between the knot complements of non-zero degree. The last condition is satisfied for all epimorphisms in [25, Theorem 1.1]. In fact, they are induced by a very nice map , called a branched-fold map [25, Theorem 1.2]. Thus it would be natural to expect that any epimorphism between -bridge knot groups is equivalent to one in [25, Theorem 1.1]. In fact, some evidence for this conjecture was provided recently by Hoste and Shanahan .
Question 1 can be regarded as a special case of the more general question that, for a given link and a bridge sphere for , which essential simple loops on are null-homotopic in . The latter question in turn can be regarded as a variation of the question that, for a given -manifold and its Heegaard surface , which essential simple loops on are null-homotopic in . In [7, Question 5.4], Minsky refined this to a certain question which generalizes Question 1. Thus our result may be regarded as an answer to a special variation of Minsky’s question (see Section 8).
The authors would like to thank Norbert A’Campo, Hirotaka Akiyoshi, Brian Bowditch, Danny Calegari, Max Forester, Koji Fujiwara, Yair Minsky, Ser Peow Tan and Caroline Series for stimulating conversations. They also thank the referee for his/her careful reading of the manuscript.
2. Main result
Consider the discrete group, , of isometries of the Euclidean plane generated by the -rotations around the points in the lattice . Set and call it the Conway sphere. Then is homeomorphic to the 2-sphere, and consists of four points in . We also call the Conway sphere. Let be the complementary 4-times punctured sphere. For each , let be the simple loop in obtained as the projection of a line in of slope . Then is essential in , i.e., it does not bound a disk in and is not homotopic to a loop around a puncture. Conversely, any essential simple loop in is isotopic to for a unique . Then is called the slope of the simple loop. Similarly, any simple arc in joining two different points in such that is isotopic to the image of a line in of some slope which intersects . We call the slope of .
A trivial tangle is a pair , where is a 3-ball and is a union of two arcs properly embedded in which is parallel to a union of two mutually disjoint arcs in . Let be the simple unknotted arc in joining the two components of as illustrated in Figure 1. We call it the core tunnel of the trivial tangle. Pick a base point in , and let be the generating pair of the fundamental group each of which is represented by a based loop consisting of a small peripheral simple loop around a component of and a subarc of joining the circle to . For any base point , the generating pair of corresponding to the generating pair of via a path joining to is denoted by the same symbol. The pair is unique up to (i) reversal of the order, (ii) replacement of one of the members with its inverse, and (iii) simultaneous conjugation. We call the equivalence class of the meridian pair of the fundamental group .
By a rational tangle, we mean a trivial tangle which is endowed with a homeomorphism from to . Through the homeomorphism we identify the boundary of a rational tangle with the Conway sphere. Thus the slope of an essential simple loop in is defined. We define the slope of a rational tangle to be the slope of an essential loop on which bounds a disk in separating the components of . (Such a loop is unique up to isotopy on and is called a meridian of the rational tangle.) We denote a rational tangle of slope by . By van Kampen’s theorem, the fundamental group is identified with the quotient , where denotes the normal closure.
For each , the 2-bridge link of slope is defined to be the sum of the rational tangles of slopes and , namely, is obtained from and by identifying their boundaries through the identity map on the Conway sphere . (Recall that the boundaries of rational tangles are identified with the Conway sphere.) has one or two components according as the denominator of is odd or even. We call and , respectively, the upper tangle and lower tangle of the 2-bridge link. The 2-bridge links are classified by the following theorem of Schubert  (cf. [3, 13]).
Theorem 2.1 (Schubert).
Two 2-bridge links and are equivalent (i.e., there is a homeomorphism from to itself sending to ), if and only if the following conditions hold.
Either or .
Let be the Farey tessellation, that is, the tessellation of the upper half space by ideal triangles which are obtained from the ideal triangle with the ideal vertices by repeated reflection in the edges. Then is identified with the set of the ideal vertices of . For each , let be the group of automorphisms of generated by reflections in the edges of with an endpoint . It should be noted that is isomorphic to the infinite dihedral group and the region bounded by two adjacent edges of with an endpoint is a fundamental domain for the action of on , by virtue of Poincare’s fundamental polyhedron theorem (see, for example, ). Let be the group generated by and . When , is equal to the free product , having a fundamental domain shown in Figure 2. Otherwise, is the group generated by reflections in the edges of or according as or . It should be noted that Theorem 2.1 says that two 2-bridge links and are equivalent if and only if there is an automorphism of which sends to . Thus the conjugacy class of the group in the automorphism group of is uniquely determined by the link .
We recall the following fact ([25, Proposition 4.6 and Corollary 4.7]) which describes the role of in the study of -bridge link groups.
For every 2-bridge link , the following holds. If two elements and of lie in the same -orbit, then and are homotopic in . In particular, if belongs to the orbit of or by , then is null-homotopic in .
Our main theorem says that the converse to the last statement in the above proposition is valid.
Main Theorem 2.3.
The loop is null-homotopic in if and only if belongs to the -orbit of or .
This theorem may be paraphrased as follows, with a detailed reason explained in Section 3.
Main Theorem 2.4.
There is an upper-meridian-pair-preserving epimorphism from to if and only if or belongs to the -orbit of or .
Since the if part is [25, Theorem 1.1], the heart of this theorem is the only if part.
The remainder of this paper is organized as follows. In Section 3, we introduce the so-called upper presentation of a 2-bridge link group, where is the upper meridian pair of . This upper presentation of a 2-bridge link group will be used throughout this paper. In Section 4, we define two sequences and of slope and two cyclic sequences and of slope all of which arise from the single relator of the presentation , and observe several important properties of these sequences so that we can adopt, in the succeeding sections, small cancellation theory which is one of the geometric techniques in combinatorial group theory. In Section 5, we show that the presentation , where , satisfies small cancellation conditions and . In Section 6, by applying the Curvature Formula of Lyndon and Schupp (see ) to van Kampen diagrams over , we obtain that if is null-homotopic in , where , then the cyclic word contains some particular part of the the cyclic word . In Section 7, we prove the only if part of Main Theorem 2.3 by showing that if a rational number belongs to a natural fundamental domain of the action of on the domain of discontinuity of , then is not null-homotopic in . In the final section, Section 8, we describe the relation of Main Theorem 2.3 with the question raised by Minsky in [7, Question 5.4].
3. Presentations of 2-bridge link groups
In this section, we introduce the upper presentation of a 2-bridge link group which we shall use throughout this paper. By van Kampen’s theorem, the link group is identified with . We call the image in the link group of the meridian pair of the fundamental group (resp. the upper meridian pair (resp. lower meridian pair). The link group is regarded as the quotient of the rank 2 free group, , by the normal closure of . This gives a one-relator presentation of the link group, which is called the upper presentation (see ).
To find the upper presentation of explicitly, let and , respectively, be the elements of represented by the oriented loops and based on as illustrated in Figure 3. Then forms the meridian pair of , which is identified with the free group . Note that intersects the disk, , in bounded by a component of and the essential arc, , on of slope , in Figure 3. Obtain a word in by reading the intersection of the (suitably oriented) loop with , where a positive intersection with (resp. ) corresponds to (resp. ). Then the cyclic word represents the free homotopy class of (see Section 4 for the precise definition of a cyclic word). It then follows that
If , then intersects and alternately, and hence and appear in alternately. It is known that there is a nice formula to find as follows (see [28, Proposition 1]).
Let and be relatively prime positive integers such that . For , let
where is the greatest integer not exceeding .
If is odd, then
If is even, then
(1) The word is obtained from the open line-segment of slope extending from to by “reading” its intersection with the vertical lattice lines (see Figure 4). The open line-segment cuts the vertical lattice line at the point with height . Note that is the height of the integer lattice point just beneath . Each time the line passes through another horizontal lattice line, the signs of the ’s change. Similarly, the word can be read from the closed line-segment which is obtained by slightly shifting the closed line-segment of slope joining with to the upper-left direction (cf. Proof of Lemma 4.7).
(2) For and , we have and .
(1) Let be the automorphism of the free group which sends the generating pair to , or . Then is conjugate to or for any .
(2) Let be the automorphism of the free group which sends the generating pair to , , or . Then is conjugate to or for any .
(1) Observe that admits a natural -action, whose generators induce the automorphisms of sending to and , respectively. Moreover, the action preserves the isotopy class of the (unoriented) loop for every . Since any automorphism satisfying the assumption is induced by an element of the -action, we obtain the desired result.
(2) Let be an automorphism of satisfying the assumption. Then it is a composition of an automorphism in (1) and the automorphism, , sending to . Observe that is induced by the half-Dehn twist along the meridian disk of and that the half-Denn twist maps to . Hence we see . This, together with (1), implies the desired result. ∎
The if part is essentially equivalent to [25, Theorem 1.1] and is proved as follows. If belongs to the -orbit of or , then Main Theorem 2.3 implies that in . Thus there is an epimorphism from to which sends the upper-meridian-pair of to the upper-meridian-pair of . To prove the remaining case, note that there is a homeomorphism preserving the upper/lower tangles, such that the restriction of to is a half-Dehn twist. Thus induces an isomorphism from to which sends the upper-meridian-pair of to the upper-meridian-pair of . So, if belongs to the -orbit of or , then we have an epimorphism sending to .
Next, we prove the only if part. Suppose that there is an upper-meridian-pair preserving epimorphism from to . Then lifts to an automorphism of the free group satisfying the condition in Lemma 3.2, modulo post composition of an inner-automorphism. Thus is conjugate to , , or by Lemma 3.2. Since is a lift of the homomorphism , or represents the trivial element of , accordingly. Hence, by Main Theorem 2.3, we see that or belongs to the -orbit of or , accordingly. ∎
4. Sequences associated with 2-bridge links
In this section, we define two sequences and of slope and two cyclic sequences and of slope all of which arise from the single relator of the presentation given in Section 3, and observe several important properties of these sequences, so that we can adopt small cancellation theory in the succeeding sections.
We first fix some definitions and notation. Let be a set. By a word in , we mean a finite sequence where and . Here we call the -th letter of the word. For two words in , by we denote the visual equality of and , meaning that if and (; ), then and and for each . For example, two words and () are not visually equal, though they are equal as elements of the free group with basis . The length of a word is denoted by . A word in is said to be reduced if does not contain or for any . A word is called cyclically reduced if all its cyclic permutations are reduced. A cyclic word is defined to be the set of all cyclic permutations of a cyclically reduced word. By we denote the cyclic word associated with a cyclically reduced word . Also by we mean the visual equality of two cyclic words and . In fact, if and only if is visually a cyclic shift of .
(1) Let be a reduced word in . Decompose into
where, for each , all letters in have positive (resp. negative) exponents, and all letters in have negative (resp. positive) exponents. Then the sequence of positive integers is called the -sequence of .
(2) Let be a cyclic word in . Decompose into
where all letters in have positive (resp. negative) exponents, and all letters in have negative (resp. positive) exponents (taking subindices modulo ). Then the cyclic sequence of positive integers is called the cyclic -sequence of . Here the double parentheses denote that the sequence is considered modulo cyclic permutations.
(3) A reduced word in is said to be alternating if and appear in alternately, i.e., neither nor appears in . A cyclic word is said to be alternating if all cyclic permutations of are alternating. In the latter case, we also say that is cyclically alternating.
The following proposition is obvious from the definition.
(1) An alternating word in is completely determined by the initial letter and the associated -sequence.
(2) Let be a cyclically reduced word in of length . Then the -sequence represents the cyclic -sequence of if and only if the initial exponent of is different from the terminal exponent of .
For a rational number with , let be the word in defined in Lemma 3.1. Then the symbol (resp. ) denotes the -sequence of (resp. cyclic -sequence of ), which is called the S-sequence of slope (resp. the cyclic S-sequence of slope ).
We shall first state Propositions 4.2, 4.3, 4.4 and 4.5 below concerning the sequences defined in the above, and then prove the propositions in the remainder of this section. Propositions 4.4 and 4.5 play crucial roles in the proof of Main Theorem 2.3. Though we need those propositions only for the sequences and with , we need to extend the definitions of and to an arbitrary positive rational number (Definition 4), in order to prove these propositions. Thus Propositions 4.2, 4.3, 4.4 and 4.5 below should be regarded as propositions for every positive rational number .
Throughout the remainder of this section, denotes a positive rational number, where and are relatively prime positive integers. Then has a continued fraction expansion
where , , and unless . Note that if , whereas if .
For the positive rational number , the sequence has length , and it represents the cyclic sequence . Moreover the cyclic sequence is invariant by the half-rotation; that is, if denotes the -th term of (), then for every integer ().
For the positive rational number , putting , we have the following.
Suppose , i.e., . Then .
Suppose . Then each term of is either or , and begins with and ends with . Moreover, the following hold.
If , then no two consecutive terms of can be , so there is a sequence of positive integers such that
Here, the symbol “” represents successive ’s.
If , then no two consecutive terms of can be , so there is a sequence of positive integers such that
Here, the symbol “” represents successive ’s.
In , Hirasawa and Murasugi defined, as one of the key notions of their paper, the sequence of signs for a pair , which actually gives rise to our -sequence of slope . They also observed several properties for the sequence of signs for , which are very similar to the properties of stated in Proposition 4.3.
If , the symbol denotes the sequence in Proposition 4.3, which is called the -sequence of slope . The symbol denotes the cyclic sequence represented by , which is called the cyclic -sequence of slope .
(1) Let . By Lemma 3.1, we see that the -sequence of is
By the formula for in Lemma 3.1, this implies
So and .
For the rational number , let be the rational number defined as
Then we have
where denotes the sequence obtained from reversing its order.
For the positive rational number , putting , the sequence has a decomposition which satisfies the following.
Each is symmetric, i.e., the sequence obtained from by reversing the order is equal to . (Here, is empty if .)
Each occurs only twice in the cyclic sequence .
begins and ends with .
begins and ends with .
is symmetric, i.e., the cyclic sequence obtained from by reversing its cyclic order is equivalent to (as a cyclic sequence). In particular, in Proposition 4.4, we actually have
(1) Let . Recall from Example 1 that
Putting and , we have
where and satisfy all the assertions in Proposition 4.5.
The remainder of this section is devoted to the proof of the propositions. We first prepare a few symbols. For a real number , let be the greatest integer not exceeding , the greatest integer smaller than , and be the smallest integer greater than . Then, for a non-integral real number , whereas for an integer . We also note that and for every and . By using this symbol, we have the following formula for the relator in the group presentation of given in Section 3.
For the positive rational number , the word is given by the following formula:
where . In particular, is alternating and cyclically reduced.
To prove Lemma 4.7, let be the line in of slope passing through the origin, and let be the line obtained by translating by the vector for sufficiently small positive real number . Then lies in and projects to the simple loop . Pick a base point, , from the intersection of with the second quadrant, and consider the sub-line-segment of bounded by and . Then it forms a fundamental domain of the covering , and the word is obtained by reading the intersection of the line-segment with the vertical lattice lines. To be precise, for each integer , let be the intersection of the line-segment with the vertical lattice line . We define the letter at to be or according as lies on a vertical edge with a single arrow or double arrow in Figure 4, namely according as is even or odd. We define the sign of to be or according as the corresponding arrow is upward or downward. Then the letter and the sign of , respectively, give the letter and the exponent of the -th term of the word for each . To describe the sign of , note that the -coordinate of is equal to , where is a sufficiently small positive real. Thus it is contained in the open interval . Thus the corresponding arrow is upward or downward according as is even or odd. Hence the sign of is equal to . This means that the exponent, , of the -th term of is . Thus we obtain the first assertion of Lemma 4.7. The second assertion is a direct consequence of the first assertion.
If , then the sequence has length , and its -th term is given by the following formula ():
where denotes the number of elements of the set.
Suppose . Then, for each integer with , the horizontal strip contains some , namely, the right hand side of the first identity is a positive integer. By this fact and by the above geometric description of and the definition of , we see that