Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.     

Representations of Knot Groups and Twisted Alexander Polynomials

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.     

On minimal elements for a partial order of prime knots

In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL₂(C)-character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al.     

Reidemeister torsion and lens surgeries on knots in homology 3-spheres II

We introduce the norm and the order of a polynomial and of a homology lens space. We calculate the norm of the cyclotomic polynomials, and apply it to lens surgery problem for a knot whose Alexander polynomial is the same as an iterated torus knot.     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Realizing Alexander polynomials by hyperbolic links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

The complete splitting number of a lassoed link

In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s−1, and less than or equal to r+split(K). In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with split(L)=r. Moreover, we construct a link L whose unlinking number is greater than split(L).     

Jones polynomial of knots formed by repeated tangle replacement operations

In this paper, we prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most k crossings (where k is a constant independent of the total number n of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by O(nk). In particular, we show that the Jones polynomial of any Conway algebraic link diagram with n crossings can be computed in O(n²) time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of n crossings can be computed in O(n²) time.     

The Thurston norm, fibered manifolds and twisted Alexander polynomials

Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita-Terasaka knot which have trivial Alexander polynomial.We also give obstructions to fibering 3-manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some of these it was unknown whether or not they are fibered. Our work in particular extends the fibering obstructions of Cha to the case of closed manifolds.     

Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

     

Representations of small braid groups in Temperley-Lieb algebra

In this paper we consider the question of faithfulness of the Jones' representation of braid group Bn into the Temperley-Lieb algebra TLn. The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493-505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B₆ into TL9,2—to the authors' knowledge the first such examples in the second gradation of the Temperley-Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.     

On zeros of the Alexander polynomial of an alternating knot

We prove that for any zero α of the Alexander polynomial of a two-bridge knot, −3<Re(α)<6. Furthermore, for a large class of two-bridge knots we prove −1<Re(α).     

Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial

We give a geometric proof of the following result of Juhasz. Let a _g be the leading coefficient of the Alexander polynomial of an alternating knot K. If |a _g | <  4 then K has a unique minimal genus Seifert surface. In doing so, we are able to generalise the result, replacing ‘minimal genus’ with ‘incompressible’ and ‘alternating’ with ‘homogeneous’. We also examine the implications of our proof for alternating links in general.     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

Nonlinear equivariant automorphisms

We show that the natural representation of SL₃ × SL₅ × SL₁₃ allows nonlinear equivariant automorphisms; more exactly, the group of polynomial automorphisms on ?³ ? ?⁵ ? ?¹³ commuting with the simple SL₃ × SL₅ × SL₁₃-action is isomorphic to ? ? ?. This is the first example of a simple module with nonlinear equivariant automorphisms.     

Twisted exponential sums of polynomials in one variable

The twisted T-adic exponential sums associated to a polynomial in one variable are studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums. This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.     

Nontrivial Alexander polynomials of knots and links

In this paper we present a sequence of link invariants, definedfrom twisted Alexander polynomials, and discuss their effectivenessin distinguishing knots. In particular, we recast and extendby geometric means a recent result of Silver and Williams onthe nontriviality of twisted Alexander polynomials for nontrivialknots. Furthermore we prove that these invariants decide ifa genus one knot is fibered. Finally we also show that theseinvariants distinguish all mutants with up to 12 crossings.     

Deformations of Symplectic Cohomology and Exact Lagrangians in ALE Spaces

ALE spaces are the simply connected hyperkähler manifolds which at infinity look like ${\mathbb{C}^{2}/G}ALE spaces are the simply connected hyperk?hler manifolds which at infinity look like \mathbbC²/G{\mathbb{C}^{2}/G}, for any finite subgroup G ì SL₂(\mathbbC){G \subset SL_2(\mathbb{C})}. We prove that all exact Lagrangians inside ALE spaces must be spheres. The proof relies on showing the vanishing of a twisted version of symplectic cohomology.