The braid alternation number

Mar´ıa de los Angeles Guevara Hernandez´ (OCAMI)

Abstract We introduce a numerical invariant that measures how far a is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by Kawauchi. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this in- variant for some infinite families. We also estimate it for with nine crossings or less.

1 Introduction

There exist numerical invariants that measure how far is a link from alternating links. In particular, Kawauchi introduced the alternation number of a link, [8]. The alternation number of a link diagram D is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating link. The alternation number of a link L, denoted by alt(L), is the minimum alternation number of any diagram of L. The alternation number of L is also the Gordian distance from L to the set of alternating links. Another numerical invariant was introduced by Adams et al [2]. The dealternating num- ber of a link diagram D is the minimum number of crossing changes necessary to transform D into an alternating diagram. The dealternating number of a link L, denoted by dalt(L), is the minimum dealternating number of any diagram of L. A link with dealternating number k is also called k-almost alternating. We say that a link is almost alternating if it is 1-almost alternating. It is immediate from their definitions that alt(L) ≤ dalt(L) for any link L. By Alexander’s theorem is known that any link can be presented as closed braids [3]; however, there are alternating links that cannot be presented as alternating braids [5]. Then a natural question is the following, what is the minimal number of crossings changes needed to transform a link into an alternating closed braid? In order to answer this question, for a link L we introduce the braid alternation and the braid dealternating number, which are link invariants that measure how far is a link from being an alternating closed braid and are denoted by Balt(L) and dBalt(L), respectively. The precise definitions are given in Section 3. Furthermore, we show that these invariants are independent of the classic definitions for links. We study the relation between these invariants showing that all of them are distinct. The value of these invariants has been obtained for some knot families. This paper is organized as follows: In Section 2, we remark some definitions concerning braids and links, and the relation between them. After that, in Section 3, the braid alternation number and the braid dealternating number are defined. In Section 4, we give a table with the braid alternation and braid dealternating numbers of prime knots up to 9 crossings.

MSC 2010: 57M25, 57M57. Key words and phrases: Alternating knot, closed braid, alternating distance, . 2 Braids and links

A braid of n-strands is an element of the n- Bn which can be expressed as a word in generators σ1,··· ,σn−1 where σi is the braid involving a single crossing of the ith and i + 1st strands. They are related by the following relations:

1. σiσ j = σ jσi, if |i − j| > 1;

2. σiσi+1σi = σi+1σiσi+1, if i = 1,...,n − 2. A braid β is said to be alternating if, and only if, it admits an alternating diagram, that is if even-numbered generators have the opposite sign to odd-numbered generators. Similarly, a link is said to be alternating if, and only if, it admits an alternating diagram. The trivial link is an example of an alternating link. We call closed braid diagrams to the diagrams of L that are presented as the closure of a braid. The unknotting number of a link L, denoted by u(L), is defined to be the minimum number of crossing changes required to convert L into the trivial link overall link diagrams representing L. There is no algorithm to compute it for a given link until now. It is known that invariants, such as the unknotting number and the alternation number can be computed in diagrams that are not presented as closed braids. However, as a consequence of Alexander’s theorem, some algorithms have surfaced for obtaining a closed braid from a given link diagram [15, 14, 9]. In particular, we note the following.

Theorem 2.1. (cf. Theorem 1.2.2 of [9]) Any link diagram D can be deformed into a closed braid diagram Bˆ such that every crossing at D remains at B.ˆ

Proof. This fact is a consequence of the proof given for Theorem 1.2.2 in [9] since it was showed that any link diagram can be deformed into a closed braid by a finite sequence of concentric deformations and the connecting arcs, which represent the crossings, remain after a concentric deformation, see Figure 1. 

(a) (b) (c) (d) (e) (f)

Figure 1: Diagrams and Seifert circles with connecting arcs before and after a concentric deformation; (a) and (d) are prior to the deformation. The original crossing and the corre- sponding connecting arcs are highlighted in blue.

Then, if the condition “all link diagrams” on the definition is restricted to “all closed braid diagrams” we can still obtain the unknotting and alternation numbers. So, we can estimate invariants such as the unknotting number and the alternation number over the set of closed braids diagrams instead of considering all the diagrams. Each link can be presented by diagrams with a distinct number of crossings. A crossing is reducible if there is a circle in the projection plane meeting the diagram transversely at that crossing, but not meeting the diagram at any other point, see Figure 2. A diagram is called reduced if none of its crossings are reducible. A reduced alternating diagram of a link L has the minimal number of crossings for its diagrams. Figure 2: A reducible crossing.

In [10] a table of links enumerated by canonical order is given, in which each link is presented as a reduced closed braid.

3 Braid alternation number and braid dealternating num- ber

In order to measure the Gordian distance between a link and the set of links that can be presented as alternating closed braids, we introduce the following new numerical invariant.

Definition 1. The braid alternation number of a closed braid diagram D is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating closed braid. The braid alternation number of a link L, denoted by Balt(L), is the minimum braid alternation number of any closed braid diagram of L.

So, by definition, the alternation number is smaller than or equal to the braid alternation number. But, we affirm that the alternation number is not equal to the braid alternation number since the alternating links have alternation number zero and however some of them cannot be presented as alternating closed braids. One form to prove this fact is by using the isotopy invariant of oriented links called the Conway polynomial ∇(L) ∈ Z[z] [4], as shown in Theorem 3.1.

Theorem 3.1. [5] Let L be an alternating link. If L can be presented as a braid whose closure is an alternating diagram, and hence a braid with the minimum number of crossings, then the leading coefficient of ∇(L) is ±1.

This theorem implies that for alternating links L we have that Balt(L) 6= 0 if the leading coefficient of ∇(L) is different to ±1. In the same way, regarding fibred links, which are links whose complements are the total space of a fiber bundle over S1, we have Lemma 3.1.

Lemma 3.1. Let L be an alternating link. If L is non-fibred, then Balt(L) 6= 0.

Proof. The Conway polynomial of fibred knots has leading coefficient ±1 [13], and for the case of alternating knots, the converse holds [12]. So, if L is a non-fibred alternating knot then the leading coefficient of ∇(L) is not equal to ±1. It follows from Theorem 3.1 that the link L cannot be presented as an alternating closed braid.  In particular, the knot 52 is alternating and therefore alt(52) = 0; however, the knot 52 is non-fibred. Therefore, it has Balt(52) 6= 0. Another form to determine if the prime knots cannot be presented as alternating closed braids is by using reduced closed braid diagrams as shown in Lemma 3.2.

Lemma 3.2. Every prime alternating closed braid D is deformed into a reduced alternating closed braid. Proof. Assume an alternating closed braid has a reducible crossing point p, see Figure 3. ±1 ±1 Let p = σi . Since p is reducible, the closed braid D does not contain any other σi . Let D1 and D2 be the closed braids obtained from the closed braid D by splicing at p. Since D is prime, D1 or D2 is a closed braid of the trivial knot. Say D2 is a trivial knot diagram. Then D1 is an alternating closed braid deformed from D. By continuing this process we obtain a reduced alternating closed braid. 

Figure 3: Alternating closed braid with a reducible crossing point.

Then, if a K has a reduced non-alternating closed braid, Balt(K) 6= 0. So, since knot 52 has a reduced non-alternating closed braid diagram [10], then it is not possible to present it as an alternating closed braid. Note 3.1. The converse of Theorem 3.1 and consequently of Lemma 3.1 does not hold. An example of this fact is given by considering the knot 1060, which is a prime alternating fibered knot; however, since this knot cannot be presented as a closed alternating braid with ten crossings, see [10] and [6], then Balt(1060) 6= 0. The braid alternation number resembles the alternation number. Similarly, we define an invariant that resembles the dealternating number. Definition 2. The braid dealternating number of a close braid diagram D is the minimum number of crossing changes necessary to transform D into an alternating closed braid di- agram. The braid dealternating number of a link L, denoted by dBalt(L), is the minimum dealternating number of any close braid diagram of L.

It follows that Balt(L) ≤ dBalt(K). The braid dealternating number is not equal to deal- ternating number, as before, if we consider the alternating knot 52 we note that dalt(52) = 0 and dBalt(52) ≥ 1 since Balt(52) ≥ 1. The following relations between these numerical invariants follow from their definitions.

dalt(L) ≤ dBalt(L)

≤ ≤ (1) alt(L) ≤ Balt(L) By Lemma 3.1, there exist many knots whose alternation numbers are different from their braid alternation numbers. However, also exist many knots with the alternation number equal to the braid alternation number. The following theorem gives a family of 3-braids with the same alternation, dealternating, braid alternation, and braid dealternating number. Theorem 3.2. Let K be a knot such that is the closure of a 3-braid of the form

2n r pi qi ∆ ∏i=1 σ1 σ2 where n ≥ 0, pi,qi ≥ 2 for i = 1,2,...,r, and ∆ = σ1σ2σ1. Then we have alt(K) = Balt(K) = dalt(K) = dBalt(K) = n + r − 1. Proof. Abe and Kishimoto in [1, Thm 3.1] showed that alt(K) = dalt(K) = n + r − 1. Then Balt(K) ≥ alt(K) = n+r −1 and dBalt(K) ≥ dalt(K) = n+r −1. On the other hand, n+r li mi following their work we can write the braid as ∏i=1 σ1σ2 , where li,mi ∈ N (see Figure 4 (a)). We can deform the diagram in Figure 4 (a) to obtain another closed braid diagram as shown in Figure 4(b). Performing alternatively n + r − 1 crossing changes in the modified strand we obtain a closed alternating braid. Therefore, dBalt(K) ≤ n + r − 1 and consequently Balt(K) = dBalt(K) = n + r − 1. 

(a)

(b)

Figure 4: Obtaining an alternating closed braid

On the other hand, in [7] was given a family of knots, denoted by D, where the difference between the alternation and the dealternating number of each knot is arbitrarily large. The 2l+1 2n family was constructed by using a 3-braid of form σ2 ∆ with l +1,n ∈ N and a 3-tangle, denoted by c, as shown in Figure 5. To prove that the alternation number is one, a crossing change was performed in the 3-tangle c obtaining a of two strands. Two equivalent diagrams of a knot diagram in D after that crossing change are shown in Figure 6 (a) and (b).

Figure 5: On the left the 3-tangle c and on the right a knot diagram of D.

Theorem 3.3. [7] For all n ∈ N there exists an infinite knot family in D, such that if K ∈ D then alt(K) = 1 and dalt(K) = n.

Figure 6: A knot diagram in D after a crossing change. Similarly to the gap between the classical alternation number and the dealternating num- ber, for each positive integer n there exist a family of infinitely many hyperbolic prime knots with braid alternation number 1 and braid dealternating number greater than or equal to n, whose braid index equal to n + 3. Theorem 3.4. Let K be a knot in D. Then we have alt(K) = Balt(K) = 1 and dBalt(K) ≥ n. Proof. Theorem 3.3 implies that if K is a knot in D, then alt(K) = 1. Consequently, Balt(K) ≥ 1. We will prove that Balt(K) ≤ 1. Let D be a diagram of a knot in D, for instance, the diagram in Figure 5 (b). After a crossing change in the 3-tangle c of D, we obtain a torus knot of two strands is obtained, which can be presented as an alternating closed braid. By Theorem 2.1, we know that D can be deformed into a closed braid Bˆ whose crossings include all crossings of D, an example is shown in Figure 7. Then, Bˆ after a crossing change in the corresponding crossing performed at D yields to an alternating closed braid. Therefore, Balt(K) = 1. Besides, as the dealternating number of K is n, then due to the inequalities (1) we have dBalt(K) ≥ n. 

Figure 7: A closed braid diagram of K in D with n = 3.

Due to Theorem 2.1 and the fact that the trivial link can be presented as an alternating closed braid, the unknotting number of a link L is an upper bound of the braid alternation number. For links, we have the following relations between these numerical invariants.

dalt(L) ≤ dBalt(L)

≤ ≤ (2) alt(L) ≤ Balt(L) ≤ u(L) In summary, we have knot families, whose knots invariants have the following relations.

• alt(K) < Balt(K) on non-fibred alternating knots, see Lemma 3.1.

• dalt(K) < dBalt(K) on alternating knots K such that Balt(K) 6= 0. • Balt(K) < dBalt(K) on the knot family D, Theorem 3.4. • alt(K) = dalt(K) = Balt(K) = dBalt(K), on the knot family given in Theorem 3.2.

Note 3.2. Since for non-fibred alternating knots we have that dalt(K) < Balt(K) and the families given in [7] have Balt(K)=1 and dalt(K) = n, then dalt(L) and Balt(L) are not comparable.

The braid alternation number is subadditive.

Proposition 3.1. Let K1 and K2 be two knots then Balt(K1#K2) ≤ Balt(K1) + Balt(K2), where # denotes the connected sum of K1 and K2. Proof. Let D1 and D2 be two diagrams of K1 and K2 respectively. By Reidemeister a b moves, we can obtain D1 and D2, two diagrams that need the minimal number of crossing changes among all the diagrams of K1 and K2, to obtain alternating closed braids. Let B1 a b and B2 be the alternating closed braids obtained from D1 and D2 by a crossing changes and b crossing changes, respectively. The connected sum of K1 and K2 has a diagram D1#D2 a b that can be deformed, by several Reidemeister moves, in D1#D2. After that, through a + b crossing changes, we obtain B1#B2. Then, B1#B2 can be deformed into an alternating closed braid, see Figure 8. 

Figure 8: Alternating connected sum of two alternating closed braids.

4 Table of prime knots

In [10], prime links with braid index up to 10 are ordered as reduced closed braid di- agrams, which have the minimal number of crossings. By using this enumeration, braid presentations in [11], and Lemma 3.2 we discern prime knots K up to nine crossings such that Balt(K) = 0, see Table 1. Note that due to Lemma 3.1, we can determine the knots up to 9 crossings with Balt(K) 6= 0. But, Lemma 3.1 is not useful to determine the value of Balt(K) for all knots with 10 crossings or more. In those cases, Lemma 3.2 can be used instead. However, for links with 11 crossings or more, an enumeration of reduced closed braids is needed. For each prime knot, up to nine crossings, with Balt(K) 6= 0 we estimated an upper bound of its braid alternation number. In particular, we calculate the dBalt for all these knots. In the cases of 92, 939, and 941 the unknotting number is used instead. For the cases of 948, and 949, the braid alternation number is calculated using explicit calculations. The invariant Balt(K) has been obtained for all prime knots up to 9 crossings except for 83,95,935,938, and 941. In the remaining cases, other criteria are needed to determine whether these knots are related to knots with Balt(K) = 0 and whether Balt(K) = dBalt(K) for these knots.

Example 1. Previously, we obtained that 0 ≤ Balt(52). An upper bound of Balt(52) can be estimated by using the unknotting number or the braid dealternating number dBalt(52). In Figure 9, braids whose closures are the knot 52 are given; the braid (c) yields to be alternating after a crossing change. Then Balt(52) = dBalt(52) = 1. (a) (b) (c)

Figure 9: Braids whose closures are the knot 52. The last one after a crossing change be- comes alternating.

Knot alt Balt dBalt Knot alt Balt dBalt Knot alt Balt dBalt 31 0 0 0 815 0 1 1 922 0 0 0 41 0 0 0 816 0 0 0 923 0 1 1 51 0 0 0 817 0 0 0 924 0 0 0 52 0 1 1 818 0 0 0 925 0 1 1 61 0 1 1 819 1 1 1 926 0 0 0 62 0 0 0 820 1 1 1 927 0 0 0 63 0 0 0 821 1 1 1 928 0 0 0 71 0 0 0 91 0 0 0 929 0 0 0 72 0 1 1 92 0 1 [1,2] 930 0 0 0 73 0 1 1 93 0 1 1 931 0 0 0 74 0 1 1 94 0 1 1 932 0 0 0 75 0 1 1 95 0 [1,2] [1,2] 933 0 0 0 76 0 0 0 96 0 1 1 934 0 0 0 77 0 0 0 97 0 1 1 935 0 [1,2,3] [1,2,3] 81 0 1 1 98 0 1 1 936 0 0 0 82 0 0 0 99 0 1 1 937 0 1 1 83 0 [1,2] [1,2] 910 0 1 1 938 0 [1,2] [1,2] 84 0 1 1 911 0 0 0 939 0 1 [1,2,3] 85 0 0 0 912 0 1 1 940 0 0 0 86 0 1 1 913 0 1 1 941 0 [1,2] [1,2,3] 87 0 0 0 914 0 1 1 942 1 1 1 88 0 1 1 915 0 1 1 943 1 1 1 89 0 0 0 916 0 1 1 944 1 1 1 810 0 0 0 917 0 0 0 945 1 1 1 811 0 1 1 918 0 1 1 946 1 1 1 812 0 0 0 919 0 1 1 947 1 1 1 813 0 1 1 920 0 0 0 948 1 1 [1,2] 814 0 1 1 921 0 1 1 949 1 1 [1,2,3] Table 1: Table of prime knots up to 9 crossings with their alternation, braid alternation, and braid dealternating numbers.

5 Acknowledgment

The author would like to thank Akio Kawauchi for his advice and comments since the content in this paper was obtained from several discussions with him during an academic stay at Osaka City University. The author was supported by the Science and Technology Council of Mexico (CONACYT). References

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OCAMI, Osaka City University, Sugimoto-cho, Sumiyoshi-ku, Osaka 558-8585, Japan e-mail: [email protected]