A relation between the crossing number and the height of a knotoid. / Korablev, Philipp; Tarkaev, Vladimir.
In: Journal of Knot Theory and its Ramifications, Vol. 30, No. 6, 2150040, 05.2021.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - A relation between the crossing number and the height of a knotoid
AU - Korablev, Philipp
AU - Tarkaev, Vladimir
N1 - Publisher Copyright: © 2021 World Scientific Publishing Company.
PY - 2021/5
Y1 - 2021/5
N2 - Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.
AB - Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.
KW - bridge
KW - crossing number
KW - height of knotoid
KW - Knotoid
KW - minimal diagram
UR - http://www.scopus.com/inward/record.url?scp=85111439916&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/5d7c8230-1e32-33b9-9897-c482897708a7/
U2 - 10.1142/S0218216521500401
DO - 10.1142/S0218216521500401
M3 - Article
AN - SCOPUS:85111439916
VL - 30
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
SN - 0218-2165
IS - 6
M1 - 2150040
ER -
ID: 88872804