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A Markov stabilization preserves the unknot closure

Example

Assume ${A C}_{ω}$ . The trivial braid $e \in B_{1}$ closes to the unknot, and both stabilizations $e σ_{1}$ and $e σ_{1}^{- 1}$ in $B_{2}$ also close to the unknot; the explicit isotopies are the R1 unwinding of the added kink, one for each sign. Here the unknot is the closure of the trivial one-strand braid.

Facts & Assumptions

Given: ${A C}_{ω}$ , the trivial braid $e \in B_{1}$ , its two stabilizations $e σ_{1} = e σ_{1}^{+}$ , $e σ_{1}^{- 1} \in B_{2}$ in the sense of Markov conjugation and stabilization moves, and the closure construction of The closure of a geometric braid.

[F1]

Stabilization adjoins one new strand on the right carrying the half twist $σ_{1}^{\pm 1}$ , we use the particular ambient isotopy constructed in proof step 1.2 of Markov moves preserve the oriented closure up to isotopy, which moves the added circle and an old-strand collar to a fixed-framing ball near the axis and exhibits the signed kink. The strand insertion and literal closure have the conventions of Markov conjugation and stabilization moves and The closure of a geometric braid.

[F2]

Assume ${A C}_{ω}$ : a stabilization preserves the oriented closure up to equivalence, because the added strand differs from the trivial strand by a kink which is unwound by one Reidemeister I isotopy; both signs of the kink are covered by the two signs of the stabilization (Markov moves preserve the oriented closure up to isotopy).

[F3]

The trivial one-strand braid closes to a single circle about the axis, the round unknot, and the closure of a braid with one cycle of its endpoint permutation has one component (The closure of a geometric braid).

Verification

1.1F1F3algebra

The closures of the stabilizations. By the preliminary ambient positioning in [F1], the two fixed-framing closures have representatives which are the round unknot with respectively a positive and a negative kink. Both closures are one-component links by [F3], since the endpoint permutation of each stabilization of $e$ is the transposition of the two strands.

2.1F2step 1.1

The R1 isotopies. The explicit isotopy for the positive sign is the R1 move that pulls the kink straight inside a small ball neighbourhood of the kink, keeping the rest of the closed braid fixed; for the negative sign the mirror isotopy unrolls the opposite kink. In both cases the R1 move is realized by an ambient isotopy by [F2], so the closure of each stabilization is equivalent to the closure of the trivial braid, the unknot.

3.1F2F3step 1.1step 2.1∎

Conclusion. Both stabilizations $e σ_{1}$ and $e σ_{1}^{- 1}$ in $B_{2}$ close to the unknot, with the explicit R1 isotopies of the two signs; the example illustrates that stabilization preserves the closure in the simplest possible case.

Depends on

Used by

Conjugacy alone does not classify braid closures Counterexample

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Sources

Birman and Brendle, Braids: A Survey, Handbook of Knot Theory chapter, author manuscript; section 2.3 and the stabilization figures, printed pp. 17-19 (standard reference, not scraped)
Traczyk, A new proof of Markov's braid theorem, Banach Center Publications 42 (1998), 409-419; section 1 (standard reference, not scraped)
Ozsvath, Stipsicz and Szabo, Grid Homology for Knots and Links, AMS Surveys and Monographs 208 (2015); section 2.1, printed pp. 13-19; Appendix B.1, printed pp. 367-372 (standard reference, not scraped)