Crossing numbers of composite knots and spatial graphs

Benjamin Bode

doi:10.1016/j.topol.2018.05.001

Crossing numbers of composite knots and spatial graphs

Benjamin Bode

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Abstract

We study the minimal crossing number c(K₁#K₂) of composite knots K₁#K₂, where K₁ and K₂ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θ_K₁,K₂ ⁿ that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K₁ and the remaining n edges into K₂. We prove that for large enough n we have c(θ_K₁,K₂ ⁿ)=n(c(K₁)+c(K₂)). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K₁#K₂).

Original language	English
Pages (from-to)	33-51
Number of pages	19
Journal	Topology and its Applications
Volume	243
Early online date	9 May 2018
DOIs	https://doi.org/10.1016/j.topol.2018.05.001
Publication status	Published - 1 Jul 2018

Structured keywords

SPOCK

Keywords

Composite knots
Crossing number
Spatial graphs
Theta-curves

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10.1016/j.topol.2018.05.001

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title = "Crossing numbers of composite knots and spatial graphs",

abstract = "We study the minimal crossing number c(K1#K2) of composite knots K1#K2, where K1 and K2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θK1,K2 n that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K1 and the remaining n edges into K2. We prove that for large enough n we have c(θK1,K2 n)=n(c(K1)+c(K2)). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K1#K2).",

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T1 - Crossing numbers of composite knots and spatial graphs

AU - Bode, Benjamin

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We study the minimal crossing number c(K1#K2) of composite knots K1#K2, where K1 and K2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θK1,K2 n that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K1 and the remaining n edges into K2. We prove that for large enough n we have c(θK1,K2 n)=n(c(K1)+c(K2)). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K1#K2).

AB - We study the minimal crossing number c(K1#K2) of composite knots K1#K2, where K1 and K2 are prime, by relating it to the minimal crossing number of spatial graphs, in particular the 2n-theta-curve θK1,K2 n that results from tying n of the edges of the planar embedding of the 2n-theta-graph into K1 and the remaining n edges into K2. We prove that for large enough n we have c(θK1,K2 n)=n(c(K1)+c(K2)). We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for c(K1#K2).

KW - Composite knots

KW - Crossing number

KW - Spatial graphs

KW - Theta-curves

UR - http://www.scopus.com/inward/record.url?scp=85046817248&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1709.05118

U2 - 10.1016/j.topol.2018.05.001

DO - 10.1016/j.topol.2018.05.001

M3 - Article (Academic Journal)

AN - SCOPUS:85046817248

VL - 243

SP - 33

EP - 51

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

ER -

Crossing numbers of composite knots and spatial graphs

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