Properties of Closed 3-Braids and Braid Representations of Links by Alexander Stoimenow

Author:Alexander Stoimenow
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(5.11)

with n, m, l ≥ 0, or some of its extensions. Note that MFW will be monotonous in m, n, l, i.e. MFW(β l, m, n+1) ≥ MFW(β l, m, n ), etc. Using symmetry assume n ≥ m.

One can check already at this stage that such words are not B-adequate. So we obtain Corollary 5.1, and for the proof of Theorem 1.​3 the rest of the argument here can be replaced by the application of Theorem 4.​5. Note that B-adequacy is invariant under isotopy preserving writhe and crossing number, so any other positive 4-braid word giving the same link is not B-adequate either.

If n + l > 0, then the 2-index syllable in (5.11) must be trivial. Otherwise remove all 2! in (2! , 3) m (if any), and split a loop from the first strand, as explained after (5.4). You remain with a word in 2 and 3, which is an extension of [232323] of MFW = 3. In total thus MFW = 4. With a similar argument we see that if m + l > 0, then the 2-index syllable is trivial.

We distinguish three cases depending on whether these arguments apply or not, i.e. whether n + l > 0 and/or m + l > 0.

Case 1. Both non-triviality arguments apply. So we have a family of words β l, m, n = [123(223) n 1(223) m 21123(223) l 21] with n, m > 0 or l > 0, and their extensions, and both and are trivial.

Case 1.1. l = 0. We assumed n, m > 0, and already for n = m = 1, the word [12322312232112321], we have MFW = 4.

Case 1.2. l > 0.

Case 1.2.1. m = n = 0. These are extensions of [1231211123(223) l 21].

The two letters “2” in the left plateau cannot be doubled ([123122112322321], [122312112322321]), neither the “3” ([123132112322321]), since MFW = 4 already for l = 1.

In the remaining cases, we find that [12312112 ∗ 3[23] + 2 ∗ 1] reduce w.r.t. the number of syllables of index 3, using the simplification described in Case 2.2.2 of Section 5.1.4. (Recall that, while “2∗” in a braid word should mean at least one letter “2”, the term “[23]+” should mean a possibly empty sequence of letters “2” and “3”. We distinguish braid words from index sequences by not putting commas between the numbers.)

Case 1.2.2. m + n > 0; this reduces to the case of m = n = 0 with some of the twos or the three in the left plateau doubled, where we found MFW = 4.

Case 2. In the case one of the non-triviality conditions on and does not apply, we have [123(223) n 12112321] with n > 0 and its extensions. (Now the right plateau is a mountain; the case m > 0, n = 0 is symmetric.)

Since MFW([123223122112321]) = 4, the right 2-index syllable in the left plateau ( in (5.11)) must be trivial.

Also, the right “2” and the “3” in the right plateau must be trivial: [1232231211123221] and [1232231211123321] have MFW = 4. (But the left “2” of the left and right plateau may not be trivial.

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