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On Markov’s theorem. (English) Zbl 1059.57002

From the introduction: The main result in this paper is to give another proof of Markov’s theorem on the representation of link types by braids. We hope that our new proof will be of interest because it gives new insight into the geometry, as follows. During recent years a fairly clear picture has emerged of the closed braid representatives of the unknot and unlink. See the main theorem in our paper [Trans. Am. Math. Soc. 329, 585–606 (1952; Zbl 0758.57005)]. Since the operation of taking the braid connected sum of any closed braid representative \(X\) of \({\mathcal X}\) with any closed braid representative \(U\) of the unknot produces a different closed braid representative of \({\mathcal X}\), a natural question is whether this process of taking braid connected sums with copies of closed braid representatives of \({\mathcal U}\) explains all of the complications in closed braid representatives of \({\mathcal X}\). Our proof of Markov’s theorem will clarify this situation. Let \(X,X'\) be any two closed braid representatives of the same knot or link type \({\mathcal X}\). We will show that there is an isotopy taking \(X_1\) to an intermediate closed braid \(X_3\) and another isotopy from \(X_3\) to \(X_2\) such that: 1. \(X_3\) is obtained from \(X_1\) by taking the braid connected sum of \(X_1\) with some number of copies of closed braid representatives of \({\mathcal U}\). 2. The isotopy that takes \(X_3\) to \(X_2\) is a push across an embedded annulus \({\mathcal A}_2\) which is a subset of a Seifert surface \(F\) for \(X_3\). In particular \(X_2\) is a preferred longitude for \(X_3\).

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups

Citations:

Zbl 0758.57005
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References:

[1] DOI: 10.1073/pnas.9.3.93
[2] Bennequin D., Asterisque 107 pp 87– (1983)
[3] Birman J. S., Studies pp 82– (1974)
[4] DOI: 10.1142/S0218216598000176 · Zbl 0907.57006
[5] Birman J. S., Studying Links Via Closed Braids V: Closed Braid Representatives of the Unlink, Trans AMS 329 (2) pp 585– (1992) · Zbl 0758.57005
[6] Lambropoulou S., Nos. 1 pp 95– (1997)
[7] DOI: 10.1017/S0305004100064161 · Zbl 0595.57007
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