scholarly journals INVARIANTS FOR LEGENDRIAN KNOTS IN LENS SPACES

2011 ◽  
Vol 13 (01) ◽  
pp. 91-121 ◽  
Author(s):  
JOAN E. LICATA
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Riemann Surfaces ◽  
Lens Spaces ◽  
Graded Algebra ◽  
Cyclic Cover ◽  
Legendrian Knots ◽  
Differential Graded Algebra ◽  
Cyclic Group Action

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra [Formula: see text] which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S1-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250059 ◽  
Author(s):  
MICHAEL B. HENRY ◽  
DAN RUTHERFORD
Keyword(s):  
Standard Form ◽  
Geometric Interpretation ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Complex Sequence ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
Definition Of ◽  
Morse Complex ◽  
Differential Counts

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
Author(s):  
Joshua M. Sabloff
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Circle Bundle ◽  
Differential Graded Algebra ◽  
Circle Bundles ◽  
Standard Contact ◽  
Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

scholarly journals Stable homotopy refinement of quantum annular homology

2021 ◽  
Vol 157 (4) ◽  
pp. 710-769
Author(s):  
Rostislav Akhmechet ◽  
Vyacheslav Krushkal ◽  
Michael Willis
Keyword(s):  
Structural Properties ◽  
Cyclic Group ◽  
Group Action ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Stable Homotopy ◽  
Spectrum Level ◽  
Link Homology ◽  
Cyclic Group Action ◽  
Equivariant Version

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

scholarly journals Computing homology invariants of Legendrian knots

2014 ◽  
Vol 23 (11) ◽  
pp. 1450056 ◽  
Author(s):  
Emily E. Casey ◽  
Michael B. Henry
Keyword(s):  
Homology Group ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Combinatorial Interpretation ◽  
Knot Invariants ◽  
Homology Groups ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
The Many ◽  
Morse Complex

The Chekanov–Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov–Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This paper gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.

scholarly journals On the iterated Hamiltonian Floer homology

2020 ◽  
pp. 2050026
Author(s):  
Erman Çı̇nelı̇ ◽  
Viktor L. Ginzburg
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Group Action ◽  
Floer Homology ◽  
Local Homology ◽  
Lefschetz Index ◽  
Cyclic Group Action ◽  
Symplectically Aspherical Manifold ◽  
Aspherical Manifold ◽  
Symplectically Aspherical

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

Hyperovals arising from a Singer group action on H(3,q2)$\mathcal{H}(3, q^2)$, q even

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Antonio Cossidente ◽  
Oliver H. King ◽  
Giuseppe Marino
Keyword(s):  
Cyclic Group ◽  
Group Action ◽  
Singer Group ◽  
Singer Cyclic Group

AbstractThe action of a Singer cyclic group of order

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