There Are Only Finitely Many 3-Superbridge Knots

Jeon, Choon Bae; Jin, Gyo Taek

doi:10.1142/s0218216501000871

Cited by 5 publications

(10 citation statements)

References 14 publications

(24 reference statements)

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“…But, we can represent the figure-eight knot in the spaceP 6 . In fact, we have a polynomial representation t → f (t), g(t), h(t) of the figure-eight knot (4 1 knot) with degree sequence (4,5,6), where A mathematica plot of this representation is shown in the following figure: By proposition 2.5, it follows that the polynomial degree of the figure-eight knot is 6. Note that in the polynomial representation of this knot, the degrees 4 and 5 of the first and second components are minimal in the sense that there is no polynomial representation of the figure-eight knot belonging to the space P 6 \P 6 .…”

Section: The Spacepmentioning

confidence: 99%

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Spaces of polynomial knots in low degree

Mishra

Raundal

2015

J. Knot Theory Ramifications

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We show that all knots up to 6 crossings can be represented by polynomial knots of degree at most 7, among which except for 5 2 , 5 * 2 , 6 1 , 6 * 1 , 6 2 , 6 * 2 and 6 3 all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any 5 crossing knot in degree 6? In this paper we try to partially answer this question. For an integer d ≥ 2, we define a setP d to be the set of all polynomial knots given by t → f (t), g(t), h(t) such that deg(f ) = d − 2, deg(g) = d − 1 and deg(h) = d. This set can be identified with a subset of R 3d and thus it is equipped with the natural topology which comes from the usual topology R 3d . In this paper we determine a lower bound on the number of path components ofP d for d ≤ 7. We define path equivalence between polynomial knots in the spaceP d and show that path equivalence is stronger than the topological equivalence.

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Section: The Spacepmentioning

confidence: 99%

“…, e 6 ) be a pattern such that this together with the projection t → f (t), g(t) describe the knot 5 2 , where e i is either 1 or −1 according to which the i th crossing is under crossing or over crossing. Let U e be a set of elements (a 0 , a 1 , a 2 , a 3 , a 4…”

Section: Introductionmentioning

confidence: 99%

Spaces of polynomial knots in low degree

Mishra

Raundal

2015

J. Knot Theory Ramifications

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“…This section is a summary of the authors' proof in [5] that 3-superbridge knots are among the 2-bridge knots up to nine crossings other than the three torus knots 5 1 , 7 1 and 9 1 .…”

Section: A Rough Census Of 3-superbridge Knotsmentioning

confidence: 99%

“…In Section 2, we summarize the authors' previous work which proved that the 3-superbridge knots are among the 2-bridge knots up to nine crossings other than the three torus knots of type (2,5), (2,7) and (2, 9) which are usually denoted by 5 1 , 7 1 and 9 1 , respectively, following the tables in [1,13].…”

Section: Introductionmentioning

confidence: 99%