“…These two questions give a sense of the degree of difficulty in describing the knot theory of equilateral polygons in 3-space beyond those having 8 or fewer edges. For these fundamental cases, one knows more about the nature and structure on knotting [CM98,Cal98,Cal01a,Cal01b].…”
Section: Knots Ephemeral Knots and Slipknotsmentioning
The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot.
“…These two questions give a sense of the degree of difficulty in describing the knot theory of equilateral polygons in 3-space beyond those having 8 or fewer edges. For these fundamental cases, one knows more about the nature and structure on knotting [CM98,Cal98,Cal01a,Cal01b].…”
Section: Knots Ephemeral Knots and Slipknotsmentioning
The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot.
“…The p [K] in the fourth column of Table W is the polygon index of K which is the minimal number of straight edges needed to form a polygonal knot equivalent to K. The values and the estimates of polygon index in the table were collected from various articles [2,3,6,10]. Roughly speaking, any local extrema can occur only at vertices and hence no more than one half of the vertices can attain local maxima.…”
Section: Primes Knots Up To Nine Crossingsmentioning
Abstract. We show that the list {3 1 , 4 1 , 5 2 , 6 1 , 6 2 , 6 3 , 7 2 , 7 3 , 7 4 , 8 4 , 8 7 , 8 9 } contains all 3-superbridge knots. We also supply the best known estimates of the superbridge index for all prime knots up to nine crossings.
“…The number or the upper limit of the range of numbers in the second column of the table is the largest integer not exceeding one half of the minimal edge number or the best-known minimal edge number [2,3,7,11]. For the five knots, 3 1 , 4 1 , 5 1 , 5 2 and 8 19 , this number is equal to the harmonic degree [18].…”
Abstract. Although there are infinitely many knots with superbridge index n for every even integer n ≥ 4, there are only finitely many knots with superbridge index 3.
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