Planar and Spherical Stick Indices of Knots

Adams, Colin; Collins, Dan; Hawkins, Katherine; Sia, Charmaine; Silversmith, Rob; Tshishiku, Bena

doi:10.1142/s0218216511008954

Cited by 6 publications

(4 citation statements)

References 9 publications

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“…There are two comments worth making here. First, Colin Adams et al [2,3] have looked at the projection stick index of a knot or link. This is the minimum number of sticks needed for a polygonal diagram of K. (For example, the projection stick number of the trefoil is 5, while the usual stick number is 6.)…”

Section: Discussionmentioning

confidence: 99%

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Ribbonlength of families of folded ribbon knots

Denne¹,

Haden²,

Troy³

et al. 2020

Preprint

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We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, (2, p) torus, twist, and pretzel knots, and these upper bounds turn out to be linear in crossing number. We give a new way to fold (p, q) torus knots, and show that their folded ribbonlength is bounded above by p + q. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 5. We then show that any (p, q) torus knot K has a constant c > 0, such that the folded ribbonlength is bounded above by c • Cr(K) 1/2 , providing an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.

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Section: Discussionmentioning

confidence: 99%

“…The first author [11] then used results of [17] to prove that any knot or link type K contains a folded ribbon knot K w such that (3) Rib(K w ) ≤ 72Cr(K) 3/2 + 32Cr(K) + 12 Cr(K) + 4.…”

Section: Introductionmentioning

confidence: 99%

Ribbonlength of families of folded ribbon knots

Denne¹,

Haden²,

Troy³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…We can define the projection stick index to be the minimum number of sticks needed for a polygonal knot diagram of K. We have just seen the unknot has projection stick index two, and regular stick index three. Together with undergraduate students, C. Adams [2,3] showed that the projection stick index of the trefoil knot is five, while the regular stick index is six (illustrated in Figure 2). Indeed, we expect the projection stick index to be smaller than the stick index since the edges in the knot diagram are not rigid sticks in space, they have crossing information instead.…”

Section: Modeling Folded Ribbon Knotsmentioning

confidence: 99%

Folded ribbon knots in the plane

Denne

2018

Preprint

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This survey reviews Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the most part, these are upper bounds of ribbonlength of twist knots and certain families of torus knots. We discuss result of G. Tian [37], which give upper bounds of ribbonlength in terms of crossing number. In addition, it turns out the choice of fold affects the ribbonlength. We end with a discussion of three different types of folded ribbon equivalence and give examples illustrating their relationship to ribbonlength.

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“…The projection stick number is the least number of line segments in any projection of a polygonal embedding of a knot. Together with undergraduate students, Colin Adams has given some results about the projection stick index of knots in [2,3]. For example, the projection stick index of the trefoil knot is five.…”

Section: Projection Stick Index and Ribbonlengthmentioning

confidence: 99%

Ribbonlength of folded ribbon unknots in the plane

Denne¹,

Mary²,

Terry³

et al. 2017

Contemporary Mathematics

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Abstract. We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of n cot(π/n) for the ribbonlength of n-stick unknots. We prove that the minimum ribbonlength for a 3-stick unknot with the same type of fold at each vertex is 3 √ 3, and such a minimizer is an equilateral triangle. We end the paper with a discussion of projection stick number and ribbonlength.

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Planar and Spherical Stick Indices of Knots

Cited by 6 publications

References 9 publications

Ribbonlength of families of folded ribbon knots

Ribbonlength of families of folded ribbon knots

Folded ribbon knots in the plane

Ribbonlength of folded ribbon unknots in the plane

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