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 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 Rotational virtual knots and quantum link invariants

2015 ◽  
Vol 24 (13) ◽  
pp. 1541008 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Knot Theory ◽  
Link Invariant ◽  
Quantum Invariants ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Virtual Knot Theory ◽  
Quantum Link ◽  
Knots And Links

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

scholarly journals BIRACK SHADOW MODULES AND THEIR LINK INVARIANTS

2013 ◽  
Vol 22 (10) ◽  
pp. 1350056 ◽  
Author(s):  
SAM NELSON ◽  
KATIE PELLAND
Keyword(s):  
Associative Algebra ◽  
Alexander Polynomial ◽  
Link Invariants ◽  
Knots And Links

We introduce an associative algebra ℤ[X, S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of ℤ[X, S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.

scholarly journals Recent results on weakly factorial domains

2018 ◽  
Vol 20 ◽  
pp. 01001
Author(s):  
Chang Gyu Whan
Keyword(s):  
Polynomial Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Laurent Polynomial ◽  
Semigroup Ring ◽  
Prime Element ◽  
Cancellative Monoid ◽  
Factorial Domain ◽  
Laurent Polynomial Ring

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

Restrictions on Homflypt and Kauffman polynomials arising from local moves

2020 ◽  
Vol 29 (06) ◽  
pp. 2050036
Author(s):  
Sandy Ganzell ◽  
Mercedes V. Gonzalez ◽  
Chloe’ Marcum ◽  
Nina Ryalls ◽  
Mariel Santos
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Laurent Polynomial Ring

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

scholarly journals Finite domination and Novikov rings: Laurent polynomial rings in two variables

2015 ◽  
Vol 14 (04) ◽  
pp. 1550055
Author(s):  
Thomas Hüttemann ◽  
David Quinn
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Homotopy Equivalent ◽  
Polynomial Rings ◽  
Cochain Complex ◽  
Finitely Generated ◽  
Bounded Complex ◽  
Laurent Polynomial Ring ◽  
Free Modules

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

scholarly journals AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1369-1422 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Finite Number ◽  
Crossing Number ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Polynomial Coefficients ◽  
Bracket Polynomial ◽  
Knots And Links

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

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.

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