Braid-like Reidemeister moves on closed braids are braid isotopies

Read a local move at a fixed cut. Choose a cut outside the small disk of the move, and straighten its local angular foliation to a braid rectangle. The untouched part gives two fixed word contexts. A braid-like II pair contributes ${\sigma }_i{\sigma }_i^{-1}$ or ${\sigma }_i^{-1}{\sigma }_i$, so removal is inverse cancellation. For III put $s={\sigma }_i$, $t={\sigma }_{i+1}$. The positive and negative cases are $sts=tst$ and its inverse. The four remaining possible over/under depth orders give $st{s}^{-1}=t^{-1}st,s^{-1}ts=ts{t}^{-1},s{t}^{-1}{s}^{-1}=t^{-1}{s}^{-1}t,s^{-1}{t}^{-1}s=t{s}^{-1}{t}^{-1}\mathrm{.}$ The first two follow by multiplying $sts=tst$ on the appropriate left and right by $t^{-1}$ or $s^{-1}$, and the last two are their inverses. These exhaust the six orders of three distinct strand depths; the two omitted sign triples would require a cyclic strict depth order. Thus the complete words agree by [F2]. By the homomorphism and actual geometric relations of [F3], each replacement is an endpoint-fixed geometric braid isotopy, also in either fixed word context. This does not require injectivity of that homomorphism.

Planar isotopy, reading order and cyclic cut. Transport the nested circles, crossing strips and a cut with the planar isotopy. This transports their oriented cyclic orders; no crossing is created and no over/under sign changes. The read word depends only on these orders. More explicitly, cut the transported annular picture and lift its circles to parallel oriented intervals. Every crossing strip is an event involving two consecutive intervals. Events meeting a common interval have their order fixed by that interval's oriented order. Any two total readings extending these finite orders differ by successive exchanges of adjacent incomparable events: move the first event of one reading to its position in the other, noting that every event passed must be incomparable, and induct on the remaining finite list. Incomparable crossing strips use disjoint pairs of intervals, so their generator indices differ by at least two; exchanging them is precisely far commutation [F2], realized geometrically by [F3]. A cut passing one or more events changes a product $uv$ to $vu=u^{-1}(uv)u$, an explicit conjugation, and records the same closed braid with a different starting page. Consequently the end readings of a transported picture, and readings using any other admissible cut, differ only by far commutations and conjugations. The transported angular foliation supplies a family of closed braids; at the end its reading agrees with the usual nested-chain reading by the same finite-order argument. There is no assertion that a fixed page gives fixed based endpoints throughout an arbitrary planar isotopy.

Closed isotopy and conjugacy. Apply step 1.1 to every local II or III replacement and step 1.2 to the planar pieces and changes of cut. Equal-word replacements yield based geometric isotopies by [F3], hence closed isotopies by [F4]. A cyclic cut change is realized by moving the starting page around the same closed braid, so also gives an isotopy through closed braids. The finite concatenation is therefore a braid isotopy of the closed braids. Algebraically the equal-word replacements leave the element unchanged and each cyclic change conjugates it; their finite composition is a conjugation in $B_n$. This proves both conclusions directly and makes the replacement by conjugation in a Markov sequence explicit. No choice-stated converse identification of geometric and Artin braids is used.