A Computation of Superbridge Index of Knots

Jeon, Choon Bae; Jin, Gyo Taek

doi:10.1142/s0218216502001743

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…In particular, Jeon and Jin [6] conjectured that the only knots with superbridge index 3 are the trefoil knot and the figure eight knot. This conjecture would be proved if it could be shown that the trefoil knot and the figure eight knot are the only non-trivial knots with conformations such that the binotrix does not intersect itself or the anti-binotrix and the bridge map realizes a value of 4 in such a conformation.…”

Section: Discussionmentioning

confidence: 99%

Duality properties of indicatrices of knots

et al. 2011

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Abstract. The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.

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Section: Discussionmentioning

confidence: 99%

Duality properties of indicatrices of knots

et al. 2011

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“…We have proved that the spaces P 3 and P 4 are path connected, so in general, we would like to conjecture the following: Proof. Suppose e = (e 1 , e 2 , e 3 ) be an arbitrary element in the set {−1, 1} 3 . By corollary 3.7.1, an element φ ∈ A 4 given by t → (e 1 t 2 + a φ t, e 2 t 3 + b φ t, e 3 t 4 + c φ t) is an embedding if and only if 3a 2 φ + 4e 2 b φ > 0 or e 1 a 3 φ + 2e 1 e 2 a φ b φ + e 3 c φ = 0.…”

Section: By Proposition 26 It Is Easy To See That the Setmentioning

confidence: 99%

“…We conjecture that there are no 5 crossing knots in degree 6. This will be more clear once the superbridge index of all the possible 3-superbridge knots in the list of Jin and Jeon [3] is completely known. We show that all 5 crossing knots and all 6 crossing knots (including the composite knots) are realized in degree 7.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…This list of 12 knots contains all the 3-superbridge knots. Details behind the selection of 12 knots and the rejection of 35 knots will be handled in [6].…”

Section: Prime Knots Up To 9 Crossingsmentioning

confidence: 99%