scholarly journals Virtual concordance and the generalized Alexander polynomial

2021 ◽  
pp. 2150030
Author(s):  
Hans U. Boden ◽  
Micah Chrisman
Keyword(s):  
Lower Central Series ◽  
Alexander Polynomial ◽  
Central Series ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Two Component ◽  
Thickened Surface

We use the Bar-Natan [Formula: see text]-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the [Formula: see text]-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in [Formula: see text]) is functorial under concordance. In addition, we extend classical results of Chen, Milnor and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is virtually concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots.

scholarly journals On the lower central series of some virtual knot groups

2020 ◽  
Vol 29 (09) ◽  
pp. 2050065
Author(s):  
Valeriy G. Bardakov ◽  
Neha Nanda ◽  
Mikhail V. Neshchadim
Keyword(s):  
Nilpotent Group ◽  
Free Product ◽  
Direct Product ◽  
Lower Central Series ◽  
Study Groups ◽  
Central Series ◽  
Virtual Knot ◽  
Cyclic Groups ◽  
Virtual Knots ◽  
Free Product With Amalgamation

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

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.

On the virtual Alexander polynomial of periodic virtual knots

2014 ◽  
Vol 23 (07) ◽  
pp. 1460011 ◽  
Author(s):  
Joonoh Kim ◽  
Sang Youl Lee ◽  
Myoungsoo Seo
Keyword(s):  
Alexander Polynomial ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Period 2

In this paper, we give a relationship between the virtual Alexander polynomial of a periodic virtual knot and that of its factor knot. As an application, we improve Im–Lee's table of possible periods of 117 virtual knots with classical crossings ≤ 4 in Jeremy Green's table. In particular, we prove that the virtual knots 4.1 and 4.77 have the actual period 2 and no others, and the virtual knot 4.99 has the actual period 2.

scholarly journals The Alexander polynomial for virtual twist knots

2017 ◽  
Vol 26 (01) ◽  
pp. 1750007
Author(s):  
Isaac Benioff ◽  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Recursive Formula ◽  
Alexander Polynomial ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Polynomial Formula

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial [Formula: see text] (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from [Formula: see text].

scholarly journals GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM

2013 ◽  
Vol 22 (12) ◽  
pp. 1341006 ◽  
Author(s):  
VLADIMIR ALEKSANDROVICH KRASNOV ◽  
VASSILY OLEGOVICH MANTUROV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Strong Sense ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crucial Difference ◽  
Virtual Links ◽  
Two Component ◽  
The Rich ◽  
Virtual Knot Theory

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

scholarly journals POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

2013 ◽  
Vol 22 (04) ◽  
pp. 1340004 ◽  
Author(s):  
ALISSA S. CRANS ◽  
ALLISON HENRICH ◽  
SAM NELSON
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Monomial Ordering ◽  
Laurent Polynomial Ring ◽  
Future Work ◽  
Knots And Links

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

scholarly journals Alexander invariants for virtual knots

2015 ◽  
Vol 24 (03) ◽  
pp. 1550009 ◽  
Author(s):  
Hans U. Boden ◽  
Emily Dies ◽  
Anne Isabel Gaudreau ◽  
Adam Gerlings ◽  
Eric Harper ◽  
...  
Keyword(s):  
Alexander Polynomial ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Crossing Numbers ◽  
Alexander Invariants

Given a virtual knot K, we introduce a new group-valued invariant VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial HK(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial GK(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VGK → GLn(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

scholarly journals On the lower central series quotients of a graded associative algebra

2011 ◽  
Vol 328 (1) ◽  
pp. 287-300 ◽  
Author(s):  
Martina Balagović ◽  
Anirudha Balasubramanian
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Central Series

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].

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