The braid alternation number Mar´ıa de los Angeles Guevara Hernandez´ (OCAMI) Abstract We introduce a numerical invariant that measures how far a link 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 knot families. We also estimate it for knots 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, unknotting number. 2 Braids and links A braid of n-strands is an element of the n-braid group Bn which can be expressed as a word in generators s1;··· ;sn−1 where si is the braid involving a single crossing of the ith and i + 1st strands. They are related by the following relations: 1. sis j = s jsi; if ji − jj > 1; 2. sisi+1si = si+1sisi+1; if i = 1;:::;n − 2: A braid b 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) 2 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 = si . Since p is reducible, the closed braid D does not contain any other si . 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 prime knot 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.