Alexander Polynomial Research Articles

Invariants of Multi-linkoids

In this paper, we extend the definition of a knotoid to multi-linkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized Theta -graphs.

𝔤𝔩(1|1)-Alexander polynomial for 3-manifolds

As an extension of Reshetikhin and Turaev’s invariant, Costantino, Geer and Patureau-Mirand constructed 3-manifold invariants in the setting of relative [Formula: see text]-modular categories, which include both semi-simple and non-semi-simple ribbon tensor categories as examples. In this paper, we follow their method to construct a 3-manifold invariant from Viro’s [Formula: see text]-Alexander polynomial. We take lens spaces [Formula: see text] and [Formula: see text] as examples to show that this invariant can distinguish homotopy equivalent manifolds.

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

AbstractWe use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum is at least . We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

Orientable maps and polynomial invariants of free-by-cyclic groups

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism φ and use it to characterize when the homological stretch factor of φ is equal to its geometric stretch factor.

The Kauffman bracket skein module of the complement of (2,2p + 1)-torus knots via braids

In this paper we compute the Kauffman bracket skein module of the complement of (2,2p+1)-torus knots, KBSM(T(2,2p+1)c), via braids. We start by considering geometric mixed braids in S3, the closure of which are mixed links in S3 that represent links in the complement of (2,2p+1)-torus knots, T(2,2p+1)c. Using the technique of parting and combing geometric mixed braids, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group B2,n and that are followed by their “coset” part, that represents T(2,2p+1)c. In that way we show that links in T(2,2p+1)c may be pushed to the genus 2 handlebody, H2, and we establish a relation between KBSM(T(2,2p+1)c) and KBSM(H2). In particular, we show that in order to compute KBSM(T(2,2p+1)c) it suffices to consider a basis of KBSM(H2) and study the effect of combing on elements in this basis. We consider the standard basis of KBSM(H2) and we show how to treat its elements in KBSM(T(2,2p+1)c), passing through many different spanning sets for KBSM(T(2,2p+1)c). These spanning sets form the intermediate steps in order to reach at the set BT(2,2p+1)c, which, using an ordering relation and the notion of total winding, we prove that it forms a basis for KBSM(T(2,2p+1)c). Note that elements in BT(2,2p+1)c have no crossings on the level of braids, and in that sense, BT(2,2p+1)c forms a more natural basis of KBSM(T(2,2p+1)c) in our setting. We finally consider c.c.o. 3-manifolds M obtained from S3 by surgery along the trefoil knot and we discuss steps needed in order to compute the Kauffman bracket skein module of M. We first demonstrate the process described before for computing the Kauffman bracket skein module of the complement of the trefoil, KBSM(Trc), and we study the effect of braid band moves on elements in the basis of KBSM(Trc). These moves reflect isotopy in M and are similar to the second Kirby moves.The “braid” method that we propose for computing Kauffman bracket skein modules seem promising in computing KBSM of arbitrary c.c.o. 3-manifolds M. The only difficulty lies in finding the sufficient relations that reduce elements in the basis of our underlying genus g-handlebody, Hg. These relations come from combing in the case of knot complements and from combing and braid band moves for 3-manifolds obtained by surgery along a knot in S3. Our aim is to set the necessary background of this “braid” approach in order to compute Kauffman bracket skein modules of arbitrary 3-manifolds.

Vanishing of higher order Alexander‐type invariants of plane curves

AbstractThe higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

Projective rigidity and Alexander polynomials of certain nodal hypersurfaces

We present nodal algebraic hypersurfaces in the complex projective space which are projectively rigid. Defects and Alexander polynomials associated with the hypersurfaces are obtained. There are families of nodal hypersurfaces with nontrivial Alexander polynomials and nodal threefolds with projective rigidity which are potentially infinite.

The homological arrow polynomial for virtual links

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface’s skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled “whiskers” that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study [Formula: see text]-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu’s characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman–Murasugi–Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an “h-reduced” diagram [Formula: see text] is [Formula: see text].

Invariants of 4–manifolds from Khovanov–Rozansky link homology

We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.

Dispersion relations, knots polynomials, and the q -deformed harmonic oscillator

We show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot polynomials and q-deformed algebras. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot $(2,2n+1)$ arising in knot theory. Certain linear combinations of the low energy expansion coefficients of the amplitude can be bounded in terms of knot invariants. Pion S-matrix bootstrap data respects the analytic bounds so obtained. We correlate the $q$-deformed harmonic oscillator with the CSDR-knot picture. In particular, the scattering amplitude can be thought of as a $q$-averaged thermal two point function involving the $q$-deformed harmonic oscillator. The low temperature expansion coefficients are precisely the $q$-averaged Alexander knot polynomials.

The Kauffman bracket skein module of the lens spaces via unoriented braids

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.

A quantum categorification of the Alexander polynomial

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.

The finiteness conjecture for skein modules

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.

Defect and degree of the Alexander polynomial

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in q^{pm 2} of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

Some congruence of the generalized Alexander invariant for periodic virtual links

In this paper, we provide two new congruences of the generalized Alexander polynomial [Formula: see text] for periodic virtual links [Formula: see text]. We use the Yang–Baxter state model of [Formula: see text] introduced by Kauffman and Radford.

An Alexander-type invariant for doodles

We construct an Alexander-type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of the second kind. Like the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

On a theorem of Friedl and Vidussi

A theorem of Friedl and Vidussi says that any 3-manifold [Formula: see text] and any non-fibered class in [Formula: see text] there exists a representation such that the corresponding twisted Alexander polynomial is zero. However, it seems that no concrete example of such a representation is known so far. In this paper, we provide several explicit examples of non-fibered knots and their representations with zero twisted Alexander polynomial.

Milnor’s concordance invariants for knots on surfaces

Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\bar{\mu}$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

We apply Dijkgraaf-Witten invariants over semidirect products of abelian groups to show that, if the k/ℓ-surgery along a knot K results in a small Seifert 3-manifold with multiplicities a1,a2,a3, then many constraints on k,a1,a2,a3 can be read off from the Alexander polynomial of K.

Hyperbolic L-space knots and their formal semigroups

For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements.