The number of oriented rational links with a given deficiency number

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500656 ◽

2021 ◽

Author(s):

Yuanan Diao ◽

Michael Lee Finney ◽

Dawn Ray

Keyword(s):

Crossing Number ◽

Number Formula ◽

Enumeration Problem ◽

Fibonacci Sequences

Let [Formula: see text] be the set of un-oriented and rational links with crossing number [Formula: see text], a precise formula for [Formula: see text] was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] and let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] ([Formula: see text]) and deficiency [Formula: see text]. In this paper, we derive precise formulas for [Formula: see text] and [Formula: see text] for any given [Formula: see text] and [Formula: see text] and show that [Formula: see text] where [Formula: see text] is the [Formula: see text]th convolution of the convolved Fibonacci sequences.

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Stick numbers of Montesinos knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500139 ◽

2021 ◽

pp. 2150013

Author(s):

Hwa Jeong Lee ◽

Sungjong No ◽

Seungsang Oh

Keyword(s):

Upper Bound ◽

Crossing Number ◽

Number Formula ◽

Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

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On fertility of knot shadows

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500807 ◽

2020 ◽

Vol 29 (11) ◽

pp. 2050080

Author(s):

Ryo Hanaki

Keyword(s):

Crossing Number ◽

Number Formula ◽

Definition Of

A knot [Formula: see text] is a parent of a knot [Formula: see text] if there exists a minimal crossing diagram [Formula: see text] of [Formula: see text] such that a subset of the crossings of [Formula: see text] can be changed to produce a diagram of [Formula: see text]. A knot [Formula: see text] with crossing number [Formula: see text] is fertile if for any prime knot [Formula: see text] with crossing number less than [Formula: see text], [Formula: see text] is a parent of [Formula: see text]. It is known that only [Formula: see text] are fertile for knots up to 10 crossings. However it is unknown whether there exist other fertile knots. A knot shadow is a diagram without over/under information at all crossings. In this paper, we introduce a definition of fertility for knot shadows. We show that if an alternating knot [Formula: see text] is fertile then the crossing number of [Formula: see text] is less than eight.

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Amphicheirality of ribbon 2-knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500698 ◽

2020 ◽

Vol 29 (09) ◽

pp. 2050069

Author(s):

Tomoyuki Yasuda

Keyword(s):

Crossing Number ◽

Small Segment ◽

Number Formula

For any classical knot [Formula: see text], we can construct a ribbon [Formula: see text]-knot [Formula: see text] by spinning an arc removed a small segment from [Formula: see text] about [Formula: see text] in [Formula: see text]. A ribbon [Formula: see text]-knot is an embedded [Formula: see text]-sphere in [Formula: see text]. If [Formula: see text] has an [Formula: see text]-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with [Formula: see text] ribbon crossings for [Formula: see text]. Thus, we can define naturally a notion on ribbon [Formula: see text]-knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer [Formula: see text] there exists an amphicheiral ribbon [Formula: see text]-knot with the ribbon crossing number [Formula: see text].

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Remarks on Suzuki’s knot epimorphism number

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500603 ◽

2019 ◽

Vol 28 (09) ◽

pp. 1950060

Author(s):

Jim Hoste ◽

Joshua Ocana Mercado ◽

Patrick D. Shanahan

Keyword(s):

Lower Bound ◽

Partial Order ◽

Continued Fraction ◽

Crossing Number ◽

Continued Fraction Expansion ◽

Number Formula

A partial order on prime knots can be defined by declaring [Formula: see text], if there exists an epimorphism from the knot group of [Formula: see text] onto the knot group of [Formula: see text]. Suppose that [Formula: see text] is a 2-bridge knot that is strictly greater than [Formula: see text] distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of [Formula: see text] in terms of [Formula: see text]. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number [Formula: see text] which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number [Formula: see text]. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

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Stick number of spatial graphs

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517501000 ◽

2017 ◽

Vol 26 (14) ◽

pp. 1750100 ◽

Cited By ~ 1

Author(s):

Minjung Lee ◽

Sungjong No ◽

Seungsang Oh

Keyword(s):

Research Program ◽

Upper Bound ◽

Upper Bounds ◽

Crossing Number ◽

Index Formula ◽

Spatial Graph ◽

Spatial Graphs ◽

Number Formula ◽

Arc Index

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

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Lattice stick number of spatial graphs

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500487 ◽

2018 ◽

Vol 27 (08) ◽

pp. 1850048

Author(s):

Hyungkee Yoo ◽

Chaeryn Lee ◽

Seungsang Oh

Keyword(s):

Upper Bound ◽

Crossing Number ◽

Cubic Lattice ◽

Spatial Graphs ◽

Number Formula

The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number [Formula: see text] of spatial graphs [Formula: see text] with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number [Formula: see text] [Formula: see text] where [Formula: see text] has [Formula: see text] edges, [Formula: see text] vertices, [Formula: see text] cut-components, [Formula: see text] bouquet cut-components, and [Formula: see text] knot components.

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Crossing number bounds in knot mosaics

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500566 ◽

2018 ◽

Vol 27 (10) ◽

pp. 1850056 ◽

Cited By ~ 2

Author(s):

Hugh Howards ◽

Andrew Kobin

Keyword(s):

Upper Bound ◽

Crossing Number ◽

Quantum States ◽

Number Formula

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer [Formula: see text] such that the knot can be represented as a knot [Formula: see text]-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an [Formula: see text]-mosaic and any knot [Formula: see text] that is represented on the mosaic, its crossing number [Formula: see text] is bounded above by [Formula: see text] if [Formula: see text] is odd, and by [Formula: see text] if [Formula: see text] is even. In the process, we develop a useful new tool called the mosaic complement.

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On the Crossing Number of $K_{m,n}$

The Electronic Journal of Combinatorics ◽

10.37236/1748 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 4

Author(s):

Nagi H. Nahas

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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