A note on band surgery and the signature of a knot

Moore, Allison H.; Vázquez, Mariel

doi:10.1112/blms.12397

Cited by 6 publications

(8 citation statements)

References 43 publications

(87 reference statements)

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“…Moore and Vazquez [15,Corollary 3.7] show that a distance one surgery along any knot in L(n, 1) yielding L(−n, 1) with n > 1 square-free and odd exists if and only if n = 5. In fact, in the proof of [15,Corollary 3.7], the assumption that n is square-free is just used to show that any knot admitting this type of surgery must be null-homologous.…”

Section: Thus the Hat Versionmentioning

confidence: 99%

“…Apart from L(n, 1) (respectively T (2, n)) is the simplest lens space (respectively 2-bridge link) type, it is also motivated from DNA topology. In biology, circular DNA (f) Band surgery from T (2, 5) to T (2, −5) constructed by Zeković (see [27] and [15]). can be modeled as a knot or link, and torus knots or links T (2, n) are a family of DNA knots or links occurring frequently in biological experiments.…”

Section: Introductionmentioning

confidence: 99%

“…Zeković [27] found a chirally cosmetic banding for the torus knot T (2, 5) as shown in Figure 1(f). Moore and Vazquez [15,Corollary 4.5] show that for all other torus knots T (2, m) with m > 1 odd and square free, there dose not exist any chirally cosmetic banding. Livingston [14,Theorem 14], using Casson-Gordon theory, further remove the constraint that m is square free in Moore-Vazquez theorem, with the exception of m = 9.…”

Section: Introductionmentioning

confidence: 99%

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Distance one surgeries on the lens space $L(n,1)$

Yang

2021

Preprint

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In this paper, we show that the lens space L(s, 1) for s = 0 is obtained by a distance one surgery along a knot in the lens space L(n, 1) with n ≥ 5 odd only if n and s satisfy one of the following cases: (1) n ≥ 5 is any odd integer and s = ±1, n, n ± 1 or n ± 4; (2) n = 5 and s = −5; (3) n = 5 and s = −9; (4) n = 9 and s = −5. As a corollary, we prove that the torus link T (2, s) for s = 0 is obtained by a band surgery from T (2, n) with n ≥ 5 odd only if n and s are as listed above. Combined with the result of Lidman, Moore and Vazquez [12], it immediately follows that the only nontrivial torus knot T (2, n) admitting chirally cosmetic banding is T (2, 5).The key ingredient of our proof is the Heegaard Floer mapping cone formula.

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Section: Thus the Hat Versionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Distance one surgeries on the lens space $L(n,1)$

Yang

2021

Preprint

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“…Additionally, knot theory has many applications in biology, particularly in DNA topology (Adams, 1994). In Moore and Vazquez (2020), coherent band surgery, conversion of a knot into a two-component link, and noncoherent band surgery, which is similar to coherent band surgery except the orientations are not retained, is used in the study of low-dimensional topology, particularly in DNA topology.…”

Section: Introductionmentioning

confidence: 99%

Knot the Usual Suspects: Finding the Diagrammatic Representations of Physical Knots

Rickards

Bhattacharya²,

Cheng³

et al. 2021

UFJUR

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In the last few hundred years, mathematicians have been attempting to describe the topological and algebraic properties of mathematical knots. Regarding the study of knots, there exists a disconnect between examining a knot’s mathematical and physical definitions. This is due to the inherent difference in the topology of an open-ended physical knot and a closed mathematical knot. By closing the ends of a physical knot, this paper presents a method to break this discontinuity by establishing a clear relation between physical and mathematical knots. By joining the ends and applying Reidemeister moves, this paper will calculate the equivalent mathematical prime or composite knots for several commonly used physical knots. In the future, it will be possible to study the physical properties of these knots and their potential to expand the field of mathematical knot theory.

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“…Figure 1 illustrates such a smoothing. 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 .…”

Section: Introductionmentioning

confidence: 99%

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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A note on band surgery and the signature of a knot

Cited by 6 publications

References 43 publications

Distance one surgeries on the lens space $L(n,1)$

Distance one surgeries on the lens space $L(n,1)$

Knot the Usual Suspects: Finding the Diagrammatic Representations of Physical Knots

Chiral smoothings of knots

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