ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS

2009 ◽  
Vol 18 (07) ◽  
pp. 973-984 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Prime Power ◽  
Alexander Polynomial ◽  
Branched Covers ◽  
Splitting Property ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Knot Concordance ◽  
Branched Cover ◽  
Linearly Independent

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

scholarly journals Knots having the same Seifert form and primary decomposition of knot concordance

2017 ◽  
Vol 26 (14) ◽  
pp. 1750103
Author(s):  
Taehee Kim
Keyword(s):  
Infinite Family ◽  
Alexander Polynomial ◽  
Primary Decomposition ◽  
Amenable Groups ◽  
Von Neumann ◽  
Knot Concordance ◽  
Linearly Independent

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–von Neumann [Formula: see text]-invariants for amenable groups developed by Cha–Orr and polynomial splittings of metabelian [Formula: see text]-invariants.

scholarly journals Upsilon Invariants from Cyclic Branched Covers

2021 ◽  
Author(s):  
Antonio Alfieri ◽  
Daniele Celoria ◽  
András Stipsicz
Keyword(s):  
Prime Power ◽  
Branched Covers ◽  
Rational Homology ◽  
Knot Concordance ◽  
Independence Results

We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals On independence of iterated Whitehead doubles in the knot concordance group

2018 ◽  
Vol 27 (01) ◽  
pp. 1850003
Author(s):  
Kyungbae Park
Keyword(s):  
Correction Term ◽  
Khovanov Homology ◽  
Floer Theory ◽  
Knot Concordance ◽  
Torus Knot ◽  
Linearly Independent ◽  
Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

scholarly journals The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4

2015 ◽  
Vol 217 ◽  
pp. 61-94
Author(s):  
Yoshiaki Goto ◽  
Keiji Matsumoto
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Series ◽  
Integral Representations ◽  
Monodromy Representation ◽  
Homology Groups ◽  
Linearly Independent ◽  
Fundamental Systems

AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

Differences of Alexander polynomials for knots caused by a single crossing change, II

2017 ◽  
Vol 26 (14) ◽  
pp. 1750097
Author(s):  
Yasutaka Nakanishi
Keyword(s):  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Single Crossing ◽  
Previous Note

In the previous note, Okada and the author gave an approach to give a characterization of Alexander polynomials for knots which are transformed by a single crossing change into a given knot whose Alexander polynomial is monic. In this note, we give a characterization in the case of [Formula: see text], and show that the Gordian distance of [Formula: see text] and [Formula: see text] is two. We also give a characterization in the cases of more three knots.

scholarly journals ALEXANDER POLYNOMIALS OF RIBBON LINKS

2011 ◽  
Vol 20 (02) ◽  
pp. 327-331
Author(s):  
JONATHAN A. HILLMAN
Keyword(s):  
Alexander Polynomial ◽  
Simple Argument ◽  
Alexander Polynomials

We give a simple argument to show that every polynomial f(t) ∈ ℤ[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links.

scholarly journals Metabelian SL(n, ) representations of knot groups IV: twisted Alexander polynomials

2013 ◽  
Vol 156 (1) ◽  
pp. 81-97 ◽  
Author(s):  
HANS U. BODEN ◽  
STEFAN FRIEDL
Keyword(s):  
Tensor Product ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

AbstractIn this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

Homology of Branched Coverings of 3-Manifolds

1992 ◽  
Vol 44 (1) ◽  
pp. 119-134
Author(s):  
John Hempel
Keyword(s):  
Classical Result ◽  
Quantitative Information ◽  
Cyclic Cover ◽  
Branched Covers ◽  
Branched Coverings ◽  
Homology Groups

AbstractWe give a relation between the homology groups H1() and H1 (M) for a branched cyclic cover → M of arbitrary closed, oriented 3-manifolds which generalizes a classical result of Plans on covers of S3 branched over a knot and provides other quantitative information as well. We include a general "free calculus" procedure for computing homology groups of branched covers and reinterpret the results in this computational setting.

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