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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 November 1, 2026
Ribbon numbers of 12-crossing knots
The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera.
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- PAR ID:
- 10653993
- Publisher / Repository:
- Journal of Knot Theory and its Ramifications
- Date Published:
- Journal Name:
- Journal of Knot Theory and Its Ramifications
- Volume:
- 34
- Issue:
- 13
- ISSN:
- 0218-2165
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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