Knots in knots: A study of classical knot diagrams

Millett, Kenneth C.

doi:10.1142/s0218216516410133

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…Supplemented by a careful visual analysis of the remaining cases, it seemed that all prime knots contained either a trefoil or a figure-eight knot or both. The resulting data lead to the conjectures proposed in the 2009 lecture and the research reported in [6] and this note.…”

Section: Historical Perspectivessupporting

confidence: 77%

“…This data lead to the conjectures proposed in the 2009 lecture. Slavik Jablan and Ken Millett explored these conjectures from which a more precise interpretation of the conjectures was formulated as reported in [6]. In this report, the preliminary results of a computer implementation of the refined analysis of all prime knots through 16 crossings was reported.…”

Section: Introductionmentioning

confidence: 93%

“…the conjecture(s) without employing the statistics of the closures to 'the sphere at infinity.' We focused on crossing changes modifying the minimal presentation of the prime knot consistent with the focus on the structure of subsegments and proposed the following definiton [6]:…”

Section: Subknots Of a Knotmentioning

confidence: 99%

“…Interesting possible cases have included the Conway knots, 8* = 8 18 , 9* = 9 40 , and 10* = 10 123 , Figure 8. The first case, 8*, has been discussed in [6] and does contain trefoils. In Table 3 we see that 9 40 does contain trefoils but that, in Table 5, 10 123 does not contain trefoils.…”

Section: A Ubiquitous Family: Trefoils Figure-eights and Others?mentioning

confidence: 99%

“…Jablan and Millett proposed a very specific definition of the knot type of a segment in a knot presentation [6]. Thus, one is lead to ask Question: How does the subknot population depend on the closure identification method?…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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More knots in knots: A study of classical knot diagrams

Millett

Rich

2017

J. Knot Theory Ramifications

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The structure of classical minimal prime knot presentations suggests that there are often, perhaps always, subsegments that present either the trefoil or the figure-eight knot. A comprehensive study of the subknots of the minimal prime knot presentations through 15 crossings shows that this is always the case for these knot presentations. Among this set of 313, 258 prime knot presentation, there are only 547, or 0.17%, that do not contain a trefoil subknot. Thus, 99.83% of minimal prime knot presentations through 15 crossings contain trefoil subknots. We identify several infinite minimal alternating prime knot families that do not contain trefoil subknots but always contain figure-eight knots. We discuss the statistics of subknots of prime knots and, using knot presentation fingerprints, illustrate the complex character of the subknots of these classic minimal prime knot presentations. We conclude with a discussion the conjectures and open questions that have grown out of our research.

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Section: Historical Perspectivessupporting

confidence: 77%

Section: Introductionmentioning

confidence: 93%

Section: Subknots Of a Knotmentioning

confidence: 99%

Section: A Ubiquitous Family: Trefoils Figure-eights and Others?mentioning

confidence: 99%

Section: Conclusion and Open Questionsmentioning

confidence: 99%

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More knots in knots: A study of classical knot diagrams

Millett

Rich

2017

J. Knot Theory Ramifications

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Knotting and linking in macromolecules

Millett

2018

Reactive and Functional Polymers

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On the Number of Unknot Diagrams

Medina¹,

Ramírez-Alfonsín²,

Salazar³

2019

SIAM J. Discrete Math.

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Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2 n diagrams. A folklore argument shows that at least one of these 2 n diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually D has more than one unknot diagram, but it cannot yield more than 4n unknot diagrams. We improve this linear bound to a superpolynomial bound, by showing that at least 2 3 √ n of the diagrams in D are unknot. We also show that either all the diagrams in D are unknot, or there is a diagram in D that is a diagram of the trefoil knot. arXiv:1710.06470v1 [math.CO]

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Knots in knots: A study of classical knot diagrams

Cited by 4 publications

References 8 publications

More knots in knots: A study of classical knot diagrams

More knots in knots: A study of classical knot diagrams

Knotting and linking in macromolecules

On the Number of Unknot Diagrams

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