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Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots, 1-bridge braids, algebraic knots, and Lorenz knots. We prove that if a knot is twist positive, the braid index appears as the third exponent in its Alexander polynomial. We provide a few applications of this result. After observing that most known examples of L-space knots are twist positive, we prove: if K is a twist positive L-space knot, the braid index and bridge index of K agree. This allows us to provide evidence for Baker’s reinterpretation of the slice-ribbon conjecture: that every smooth concordance class contains at most one fibered, strongly quasipositive knot. In particular, we provide the first example of an infinite family of positive braid knots which are distinct in concordance, and where, as g tends to infinity, the number of hyperbolic knots of genus g gets arbitrarily large. Finally, we collect some evidence for a few new conjectures, including the following: the braid and bridge indices agree for any L-space knot.
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This content will become publicly available on May 1, 2026
The algebraic structure of hyperbolic graph braid groups
Genevois recently classified which graph braid groups are word hyperbolic. In the 3-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs: sun and pulsar graphs. We prove that 3-strand braid groups of sun graphs are free. On the other hand, it was known to experts that 3-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups.
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- PAR ID:
- 10591175
- Publisher / Repository:
- World Scientific Connect
- Date Published:
- Journal Name:
- International Journal of Algebra and Computation
- Volume:
- 35
- Issue:
- 03
- ISSN:
- 0218-1967
- Page Range / eLocation ID:
- 329 to 342
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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