scholarly journals A graphical construction of the sl(3) invariant for virtual knots

10.4171/qt/57 ◽  
2014 ◽  
Vol 5 (4) ◽  
pp. 523-539 ◽  
Author(s):  
Louis Kauffman ◽  
Vassily Manturov
Keyword(s):  
Virtual Knots ◽  
Graphical Construction

scholarly journals Minimal diagrams of free knots

2014 ◽  
Vol 23 (06) ◽  
pp. 1450032
Author(s):  
Tomas Boothby ◽  
Allison Henrich ◽  
Alexander Leaf
Keyword(s):  
Equivalence Class ◽  
Virtual Knots ◽  
Knot Diagrams

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that is irreducibly odd, then it is minimal with respect to the number of classical crossings. Not all minimal diagrams of free knots are associated to irreducibly odd graphs, however. We introduce a family of free knot diagrams that arise from certain permutations that are minimal but not irreducibly odd.

Graphical Construction of Conic Sections

1966 ◽  
Vol 4 (1) ◽  
pp. 19-19
Author(s):  
James L. Cronin
Keyword(s):  
Conic Sections ◽  
Graphical Construction

Finiteness of the set of virtual knots with a given state number

2018 ◽  
Vol 27 (08) ◽  
pp. 1850049
Author(s):  
Takuji Nakamura ◽  
Yasutaka Nakanishi ◽  
Shin Satoh
Keyword(s):  
The Real ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Number Formula

A state of a virtual knot diagram [Formula: see text] is a collection of circles obtained from [Formula: see text] by splicing all the real crossings. For each integer [Formula: see text], we denote by [Formula: see text] the number of states of [Formula: see text] with [Formula: see text] circles. The [Formula: see text]-state number [Formula: see text] of a virtual knot [Formula: see text] is the minimum number of [Formula: see text] for [Formula: see text] of [Formula: see text]. Let [Formula: see text] be the set of virtual knots [Formula: see text] with [Formula: see text] for an integer [Formula: see text]. In this paper, we study the finiteness of [Formula: see text]. We determine the finiteness of [Formula: see text] for any [Formula: see text] and [Formula: see text] for any [Formula: see text].

scholarly journals Groups of the virtual trefoil and Kishino knots

2018 ◽  
Vol 27 (13) ◽  
pp. 1842009
Author(s):  
Valeriy G. Bardakov ◽  
Yuliya A. Mikhalchishina ◽  
Mikhail V. Neshchadim
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Lower Central Series ◽  
Central Series ◽  
Virtual Link ◽  
Virtual Knots ◽  
Formal Power

In the paper [13], for an arbitrary virtual link [Formula: see text], three groups [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that [Formula: see text] distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

scholarly journals Semiquandles and flat virtual knots

2010 ◽  
Vol 248 (1) ◽  
pp. 155-170 ◽  
Author(s):  
Allison Henrich ◽  
Sam Nelson
Keyword(s):  
Virtual Knots
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