In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in $\mathbb S^3$, then $M_K$ is not commensurable to a fibered knot complement in a $\mathbb Z/ 2 \mathbb Z$-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.
We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover if two hyperbolic knots have cyclically commensurable complements, then they are fibered with the same genus and are chiral. A characterisation of cyclic commensurability classes of complements of periodic knots is also given. In the non-periodic case, we reduce the characterisat…
Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.
We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.
In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(ε, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmented links complements. We show that such knot complements have no hidden symmetries, which implies that there are at most two other knot complements in their respective commensurability classes. Under mild additional hypotheses, we show that these knots have at most four (orienta…
We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.
In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.
Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of such twisted torus knots with a single negative twist.
In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Our work requires a carefu…
The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a prev…
We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.
We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calculations. We use the Pythagorean Theorem \cite{HMP} to relate the complex length of the longitude and the complex distance between the two axes fixed by two generators. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic twist knot or…
A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.
We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in \cite{DHY}.
We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $Δ_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $Δ_0$.
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.
The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along branched twist spins are studied by Fintushel, Gordon and Pao. In this paper, we determine the 2-knot obtained from $K$ by the Gluck twist along $K^{\prime}$. As an application, we give infinitely many pairs of inequivalent branched twist spins whose complements are homeomorphic…
We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots.
In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.
For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these knots is left-orderable. There are knots, including the figure-eight knot, for which this set is empty. We give the first class of hyperbolic knots, not of this type, for which these integers can be completely determined.
A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. To prove the assertion, we give a presentation of the fundamental group of the complement of a branched twist spin, which generalizes a presentation of Plotnick, calculate the first elementary ideals and obtain the condition of the knot determinants by substituting $-1$ for t…
An $(m,n)$-branched twist spin is a fibered $2$-knot in $S^4$ which is determined by a $1$-knot $K$ and coprime integers $m$ and $n$. For a $1$-knot, Lin proved that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of a $1$-knot up to conjugation is determined by the knot determinant of the $1$-knot. In this paper, we prove that the number of irreducible $SL(2,\mathbb{C})$-metabelian representations of the knot group of an $(m,n)$-branched twist spin up to conjugation is determined by the determinant of a $1$-knot in the orbit space by comparing a prese…
We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expansion of the colored Jones polynomial.
This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.
In this paper, we show that the Gluck twist of certain satellite $2$-knots in a $4$-manifold do not change the diffeomorphism type in three different ways: one is directly from the definition of the satellite $2$-knot, and the other two are by finding an equivalent description of the satellite $2$-knot. Furthermore, using the new description, we gave infinite number of new examples of $2$-knots which are unknotted by connected summing a single standard real projective plane.
In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a slice knot $3_1 \sharp 3^*_1$. By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.
We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.
In a classic paper Zeeman introduced the k-twist spin of a knot K and showed that the exterior of a twist spin fibers over S^1. In particular this result shows that the knot K # -K is doubly slice. In this paper we give a quick proof of Zeeman's result. The k-twist spin of K also gives rise to two metabolizers for K # -K and we determine these two metabolizers precisely.
We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in terms of the distance between the two axes fixed by two generators. In this way the calculation becomes easier than using the singular locus directly. The volumes of the hyperbolic twist knot cone-manifolds simpler than Stevedore's knot are known. As an application, we give the vo…
We use a link invariant defined by Cimasoni-Florens to compute ρ-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that the twist knots of algebraic order two are linearly independent in the topological concordance group.
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high D…
It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such that $K$ is ambient isotopic to a component of $L$. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.
Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $ω$ between $K$ and $c$. We establish the conjecture in case of $|ω| \neq 1$, prove that $\{K_n\}$ contains at most three L-space knots if $ω= 0$, and address the case where $|ω| = 1$ under an additional hypothesis about Seifert surgeries. To that end, we characterize a twisting circle $c$ for which $\{ (K_n, r_n) \}$ contains at least ten S…
We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids.
Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use…
A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there i…
We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.
We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the course of proof, we provide the optimal upper bounds of the primitive finite type invariants of degree 2 and 3 that solve Willerton's conjecture.
X.S. Lin and O. Dasbach proved that the sum of the absolute value of the second and penultimate coefficients of the Jones polynomial of an alternating knot is equal to the twist number of the knot. In this paper we give a new proof of their result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.
A rational knot or link can be put into a standard alternating format which has horizontal and vertical twist sites (double helices). The number and type of these twist sites are determined by terms of next-to-highest $z$-degree in Kauffman's regular isotopy invariant $Λ(a,z)$. In particular, for a knot or link with $c$ crossings, the coefficient of the $z^{c-2}$ term is equal to the number of twist sites in its standard diagram. Furthermore, the coefficients of the $a^{-2}z^{c-2}$ and $a^2z^{c-2}$ terms count the number of left-turning and right-turning twist sites, respectively.
We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly.
In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher, Miller and Stover. The construction comes from a family of twist knot complements and their dihedral covers. The case $k=1$ arises from the uniqueness of an immersed totally geodesic thrice-punctured sphere, answering a question of Reid. Applying the proof techniques of the main result, we explicitly construct non-elementary maximal Fuchsian subgroups of infi…
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113.
A rational number $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained from $S^3$ by $r$-surgery along $K$ has left orderable fundamental group. In this paper we consider the double twist knots $C(k,l)$ in the Conway notation. For any positive integers $m$ and $n$, we show that if $K$ is a double twist knot of the form $C(2m,-2n)$, $C(2m+1, 2n)$ or $C(2m+1, -2n)$ then there is an explicit unbounded interval $I$ such that any rational number $r \in I$ is a left orderable slope of $K$.
We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichmüller TQFT and to prove the associated v…
