scholarly journals A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

2020 ◽  
Vol 29 (06) ◽  
pp. 2050037
Author(s):  
Taketo Sano
Keyword(s):  
Integral Domain ◽  
Invertible Element ◽  
Link Diagram ◽  
Link Invariant ◽  
Canonical Class

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].

A NON-ORTHOGONAL CAYLEY–DICKSON DOUBLING

2006 ◽  
Vol 05 (02) ◽  
pp. 193-199 ◽  
Author(s):  
S. PUMPLÜN
Keyword(s):  
Integral Domain ◽  
Invertible Element ◽  
Construction Method ◽  
Quotient Field ◽  
Octonion Algebras

Let R be an integral domain in which 2 is not an invertible element, with quotient field K of characteristic not 2. A construction method for octonion algebras over Ris presented for which the resulting algebra does not necessarily contain a composition subalgebra.

scholarly journals Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

scholarly journals Quantum (sln, ∧V n ) link invariant and matrix factorizations

2011 ◽  
Vol 204 ◽  
pp. 69-123
Author(s):  
Yasuyoshi Yonezawa
Keyword(s):  
Matrix Factorizations ◽  
Link Diagram ◽  
Link Invariant ◽  
Poincaré Polynomial ◽  
Oriented Link

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where∧Vnis the set of fundamental representations ofUq(sln). In the case of an oriented link diagram composed of[k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general[i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

scholarly journals A twisted link invariant derived from a virtual link invariant

2017 ◽  
Vol 26 (09) ◽  
pp. 1743007
Author(s):  
Naoko Kamada
Keyword(s):  
Knot Theory ◽  
Orientable Surface ◽  
Link Diagram ◽  
Link Invariant ◽  
Virtual Knot ◽  
Chord Diagrams ◽  
Link Diagrams ◽  
Virtual Links ◽  
Virtual Knot Theory ◽  
Double Coverings

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

scholarly journals Quantum (sln, ∧V n) link invariant and matrix factorizations

2011 ◽  
Vol 204 ◽  
pp. 69-123 ◽  
Author(s):  
Yasuyoshi Yonezawa
Keyword(s):  
Matrix Factorizations ◽  
Link Diagram ◽  
Link Invariant ◽  
Poincaré Polynomial ◽  
Oriented Link

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where ∧Vn is the set of fundamental representations of Uq(sln). In the case of an oriented link diagram composed of [k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general [i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals On complete intersections with trivial canonical class

2015 ◽  
Vol 268 ◽  
pp. 339-349
Author(s):  
Lev A. Borisov ◽  
Zhan Li
Keyword(s):  
Complete Intersections ◽  
Canonical Class
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