Stick knots

Furstenberg, Eric; Li, Jie; Schneider, Jodi

doi:10.1016/s0960-0779(97)00093-3

Cited by 28 publications

(56 citation statements)

References 5 publications

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“…If ρ meetsl i at three or more points, then the condition (4) holds. Therefore we reach the same contradiction as in (5). Suppose ρ meetsl i at two points.…”

Section: Proof Of Theoremmentioning

confidence: 74%

“…Then, for w = v ρ ′ or w = − v ρ ′ , the condition (6) holds. This leads to the same contradiction as in (5). If there are no arcs of π(K) inside R ∪ S, we can straighten a part of l i as in Figure 3(iii) without changing the knot type.…”

Section: Proof Of Theoremmentioning

confidence: 81%

“…All 3-superbridge knots are among the forty seven knots which are 2-bridge knots up to 9 crossings except the torus knots of types (2,5), (2,7) and (2,9). They are marked with ⋆ or × in Table 1.…”

Section: Prime Knots Up To 9 Crossingsmentioning

confidence: 99%

“…For the five knots, 3 1 , 4 1 , 5 1 , 5 2 and 8 19 , this number is equal to the harmonic degree [18]. It is known that 2-bridge knots cannot have superbridge index bigger than seven [5].…”

Section: Prime Knots Up To 9 Crossingsmentioning

confidence: 99%

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There Are Only Finitely Many 3-Superbridge Knots

Jeon¹,

Jin²

2001

J. Knot Theory Ramifications

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“…If ρ meetsl i at three or more points, then the condition (4) holds. Therefore we reach the same contradiction as in (5). Suppose ρ meetsl i at two points.…”

Section: Proof Of Theoremmentioning

confidence: 74%

Section: Proof Of Theoremmentioning

confidence: 81%

Section: Prime Knots Up To 9 Crossingsmentioning

confidence: 99%

“…For the five knots, 3 1 , 4 1 , 5 1 , 5 2 and 8 19 , this number is equal to the harmonic degree [18]. It is known that 2-bridge knots cannot have superbridge index bigger than seven [5].…”

Section: Prime Knots Up To 9 Crossingsmentioning

confidence: 99%

See 2 more Smart Citations

There Are Only Finitely Many 3-Superbridge Knots

Jeon¹,

Jin²

2001

J. Knot Theory Ramifications

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“…12 Además de "minimal polygon index," otros autores en Inglés se refieren a este invariante como "stick number" [1,8,10], "broken line number" [21] o simplemente "edge number" [19,24]. 13 Esta construcción aparece en la prueba del Teorema 7 de [21] y como el Ejercicio 1.38 de [1].…”

Section: Proyecciones De Nudos Y Elíndice Poligonal Mínimounclassified

Teoría de nudos geométricos e isotopía poligonal

Soto

2001

RMTA

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ResumenEl espacio de los polígonos de n lados, inmersos en el espacio euclídeo de tres dimensiones, consiste de una variedad suave en la cual los puntos corresponden a nudos lineales a trozos o "geométricos", mientras que los arcos corresponden a isotopías que preservan la estructura geométrica de esos nudos. Se describe la topología de estos espacios para los casos n = 6 y n = 7. En ambos casos, cada espacio consta de cinco componentes, aunque contiene sólo tres (cuando n = 6) o cuatro (cuando n = 7) tipos topológicos de nudos. Por lo tanto la "equivalencia geométrica de nudos" es estrictamente más fuerte que la equivalencia topológica. Este hecho se demuestra con el nudo trébol hexagonal y el nudo doble heptagonal, los cuales, a diferencia de sus contrapartes topológicas, no son reversibles. Se discutirán también las extensiones de estos resultados a los casos n ≥ 8.Palabras clave: nudos poligonales, polígonos espaciales, espacios de nudos, invariantes de nudos. AbstractThe space of n-sided polygons embedded in euclidean three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains only three (when n = 6) or four (when n = 7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n ≥ 8 will also be discussed.

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The Minimal Number of Triangles Needed to Span a Polygon Embedded in ℝd

Hass

Lagarias

2003

Algorithms and Combinatorics

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Given a closed polygon P having n edges, embedded in R d , we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P . The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P . We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n 2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2 ) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2 ) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7.

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Stick knots

Cited by 28 publications

References 5 publications

There Are Only Finitely Many 3-Superbridge Knots

There Are Only Finitely Many 3-Superbridge Knots

Teoría de nudos geométricos e isotopía poligonal

The Minimal Number of Triangles Needed to Span a Polygon Embedded in ℝd

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