Recent advances on the non-coherent band
                    surgery model for site-specific
                    recombination

Moore, Allison H.; Vázquez, Mariel

doi:10.1090/conm/746/15004

Cited by 6 publications

(7 citation statements)

References 69 publications

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“…The first knot that was shown to be algebraically slice but not slice by Casson and Gordon [2] was the two-bridge knot B (25,2). Since the appearance of a prime of even power in the first homology introduces challenges, we consider a related family of examples, the set of four two-bridge knots {B (25,1), B (25,24) In general, if there is a smoothing that converts a knot K into a knot J, then K # −J bounds a Mobius band in the four-ball. The bounds based on Theorem 7 continue to apply.…”

Section: Smoothing Distancementioning

confidence: 99%

“…In the reverse direction, Moore and Vazquez [23] showed that T (2, 5) is unique among positive torus knots T (2, m), with m square-free, for which such a move exists. The paper [24] reports on extensive computer searches that have discovered chiral smoothings for the knots 8 8 and 8 20 . If K supports a chiral smoothing, then we will see in Theorem 1 that a single band move converts K # K into K # −K, where −K denotes the mirror image of K with string orientation reversed.…”

Section: Introductionmentioning

confidence: 99%

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Chiral smoothings of knots

Livingston

2020

Proceedings of the Edinburgh Mathematical Society

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Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

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Section: Smoothing Distancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chiral smoothings of knots

Livingston

2020

Proceedings of the Edinburgh Mathematical Society

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“…In the first example, we take time to discuss what it shows in regard to a question about band surgeries on knots posed in [12]. We refer the reader to that paper and [19, Section 3] for standard definitions surrounding band surgery. Example Consider the two‐strand

- 1

‐twist on the unknot

K

shown in Figure 5.…”

Section: Two‐strand Twists Of Odd Ordermentioning

confidence: 99%

Cosmetic two‐strand twists on fibered knots

Rogers

2020

Journal of Topology

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Let K be a knot in a rational homology sphere M. This paper investigates the question of when modifying K by adding m > 0 half-twists to two oppositely oriented strands, while keeping the rest of K fixed, produces a knot isotopic to K. Such a two-strand twist of order m, as we define it, is a generalized crossing change when m is even and a non-coherent band surgery when m = ±1. A cosmetic two-strand twist on K is a non-nugatory one that produces an isotopic knot. We prove that fibered knots in M admit no cosmetic generalized crossing changes. Further, we show that if K is fibered, then a two-strand twist of odd order m that is determined by a separating arc in a fiber surface for K can only be cosmetic if m = ±1. After proving these theorems, we further investigate cosmetic two-strand twists of odd orders. Through two examples, we find that the second theorem above becomes false if 'separating' is removed, and that a key technical proposition fails when the order equals 1. A closer look at an order-one example, an instance of cosmetic band surgery on the unknot, reveals it to be nearly trivial in a sense that we name weakly nugatory. We correct the technical proposition to obtain a means of using double branched covers to show that certain band surgeries are weakly nugatory. As an application, we prove that every cosmetic band surgery on the unknot is of this type. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.

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“…Band surgery is of interest in low-dimensional topology, in particular in knot theory, and in the study of surfaces in fourmanifolds. It is of independent interest in DNA topology; for a more detailed commentary on this perspective, we refer the reader to [18,Section 5;26], which review the relevance of band surgery to the study of DNA.…”

Section: Introductionmentioning

confidence: 99%

A note on band surgery and the signature of a knot

Moore

Vázquez

2020

Bull. London Math. Soc.

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Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots K and K of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer dinvariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that T (2, 5) is the only torus knot T (2, m) with m square-free that admits a chirally cosmetic banding, that is, a band surgery operation to its mirror image. We conclude with a discussion on the scarcity of chirally cosmetic bandings.

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Recent advances on the non-coherent band surgery model for site-specific recombination

Cited by 6 publications

References 69 publications

Chiral smoothings of knots

Chiral smoothings of knots

Cosmetic two‐strand twists on fibered knots

A note on band surgery and the signature of a knot

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