Elementary Ideals Research Articles

Biquandle module invariants of oriented surface-links

We define invariants of oriented surface-links by enhancing the biquandle counting invariant using biquandle modules, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials.

Elementary Ideals in Tsemi-Rings Andn(A)-Tsemi-Ring

In this paper” we will introduce l-elementary ideals, lt-elementary ideals, r-elementary ideals and elementary ideals in TSemi-rings and its properties. Also, we will discuss special type of Tsemi-rings i.e. N(I)-TSemi-ring.

Elementary Ideals in Tsemi-Rings Andn(A)-Tsemi-Ring

In this paper” we will introduce lelementary ideals, lt-elementary ideals, r-elementary ideals and elementary ideals in TSemi-rings and its properties. Also, we will discuss special type of Tsemi-rings i.e. N(I)-TSemi-ring.

Multivariate Alexander colorings

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.].

Branched twist spins and knot determinants

A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. To prove the assertion, we give a presentation of the fundamental group of the complement of a branched twist spin, which generalizes a presentation of Plotnick, calculate the first elementary ideals and obtain the condition of the knot determinants by substituting $-1$ for the indeterminate.

Alexander invariants for virtual knots

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.

POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

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.

Levels of knotting of spatial handlebodies

If H is a spatial handlebody, i.e. a handlebody embedded in the 3-sphere, a spine of H is a graph Γ ⊂ S3 such that H is a regular neighbourhood of Γ. Usually, H is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody H to classical topological properties of its complement M = S3 \H, such as its cut number. More precisely, we show that if H is not highly knotted, then M admits special cut systems for M , and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle–coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups.

TWIN GROUPS OF VIRTUAL 2-BRIDGE KNOTS AND ALMOST CLASSICAL KNOTS

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups.

RIBBON LINK AND SEPARATE RIBBON LINK

We shall study several circles in the 3-sphere called a link which has "high" splitness properties. We offer several kind of those links and study relations among them. Alexander polynomial, Conway polynomial and Milnor μ and [Formula: see text] invariants did not work for those links as vanishing cause of high splitness. We use higher order elementary ideals to distinguish those links.

Knot modules and the elementary divisor theorem

We show that a knot module M is a cyclic module if and only if E 1( M)= Λ, all Steinitz-Fox-Smythe invariants of quotients of M by irreducible factors of Δ 0( M) are trivial, and certain unit class invariants describing the extensions in a composition series for M are trivial. (We give several counter-examples to illustrate the independence of these conditions.) We show also that a direct sum of cyclic modules (over any noetherian factorial domain) satisfies the Elementary Divisor Theorem if and only if all its elementary ideals are principal, and we replace the Dedekind condition in Levine's π-primary sequence realization theorem by a (weaker) projectivity condition.

Milnor’s invariants and the completions of link modules

Let L L be a tame link of μ ⩾ 2 \mu \geqslant 2 components in S 3 {S^3} , H H the abelianization of its group π 1 ( S 3 − L ) {\pi _1}({S^3} - L) , and I H IH the augmentation ideal of the integral group ring Z H {\mathbf {Z}}H . The I H IH -adic completions of the Alexander module and Alexander invariant of L L are shown to possess presentation matrices whose entries are given in terms of certain integers μ ( i 1 , … , i q ) \mu ({i_1}, \ldots ,{i_q}) introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of L L with each other to all be zero. In the special case μ = 2 \mu = 2 , it is shown that the various Milnor invariants μ ¯ ( [ r + 1 , s + 1 ] ) \bar \mu ([r + 1,s + 1]) are determined (up to sign) by the Alexander polynomial of L L , and that this Alexander polynomial is 0 0 iff μ ¯ ( [ r + 1 , s + 1 ] ) = 0 \bar \mu ([r + 1,s + 1]) = 0 for all r , s ⩾ 0 r,s \geqslant 0 with r + s r + s even; also, the Chen groups of L L are determined (up to isomorphism) by those nonzero μ ¯ ( [ r + 1 , s + 1 ] ) \bar \mu ([r + 1,s + 1]) with r + s r + s minimal. In contrast, it is shown by example that for μ ⩾ 3 \mu \geqslant 3 the Alexander polynomials of a link and its sublinks do not determine its Chen groups.

Linking Numbers and The Elementary Ideals of Links

Let $L = {K_1} \cup \cdots \cup {K_\mu } \subseteq {S^3}$ be a tame link of $\mu \geqslant 2$ components, and $H$ the abelianization of $G = {\pi _1}({S^3} - L)$. Let $\mathcal {L} = ({\mathcal {L}_{ij}})$ be the $\mu \times \mu$ matrix with entries in $\mathbf {Z}H$ given by $\mathcal {L}{_{ii}} = \sum \nolimits _{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)}$ and for $i \ne j {\mathcal {L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})$. Then if $0 < k < \mu$ \[ \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} } \] Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.

Linking numbers and the elementary ideals of links

Let L = K 1 ∪ ⋯ ∪ K μ ⊆ S 3 L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3} be a tame link of μ ⩾ 2 \mu \geqslant 2 components, and H H the abelianization of G = π 1 ( S 3 − L ) G = {\pi _1}({S^3} - L) . Let L = ( L i j ) \mathcal {L} = ({\mathcal {L}_{ij}}) be the μ × μ \mu \times \mu matrix with entries in Z H \mathbf {Z}H given by L i i = ∑ k ≠ i l ( K i , K k ) ⋅ ( t k − 1 ) \mathcal {L}{_{ii}} = \sum \nolimits _{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)} and for i ≠ j L i j = l ( K i , K j ) ⋅ ( 1 − t i ) i \ne j\,{\mathcal {L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i}) . Then if 0 &gt; k &gt; μ 0 &gt; k &gt; \mu \[ ∑ i = 0 k − 1 E μ − k + i ( L ) ⋅ ( I H ) 2 i + ( I H ) 2 k = ∑ i = 0 k − 1 E μ − k + i ( L ) ⋅ ( I H ) 2 i + ( I H ) 2 k \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} } \] Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.

Elementary Surgery Manifolds and the Elementary Ideals

Elementary Surgery Manifolds and the Elementary Ideals

Elementary surgery manifolds and the elementary ideals

We prove the following: If M 3 {M^3} is a closed 3-manifold obtained by elementary surgery on a knot K in S 3 {S^3} and H 1 ( M 3 ) {H_1}({M^3}) is a nontrivial cyclic group, then the first elementary ideal π 1 ( M 3 ) {\pi _1}({M^3}) in the integral group ring of H 1 ( M 3 ) {H_1}({M^3}) is the principal ideal generated by the polynomial of K.

On elementary ideals of projective planes in the 4-sphere and oriented Θ-curves in the 3-sphere

On elementary ideals of projective planes in the 4-sphere and oriented Θ-curves in the 3-sphere

On elementary ideals of𝜃-curves in the 3-sphere and 2-links in the 4-sphere

On elementary ideals of<i>𝜃</i>-curves in the 3-sphere and 2-links in the 4-sphere

On elementary ideals of polyhedra in the 3-sphere

On elementary ideals of polyhedra in the 3-sphere

On the Elementary Ideals of Link Modules

On the Elementary Ideals of Link Modules