2-bridge Knots Research Articles

Affine cubic surfaces and character varieties of knots

It is known that the fundamental group homomorphism π1(T2)→π1(S3∖K) induced by the inclusion of the boundary torus into the complement of a knot K in S3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL2-character varieties of the corresponding fundamental groups. In our earlier work [3], we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C3. In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings.

On the volume and Chern–Simons invariant for 2-bridge knot orbifolds

This paper extends the work by Mednykh and Rasskazov presented in [On the structure of the canonical fundamental set for the 2-bridge link orbifolds, Universität Bielefeld, Sonderforschungsbereich 343, Discrete Structuren in der Mathematik, Preprint (1988), pp. 98–062, www.mathematik.uni-bielefeld.de/sfb343/preprints/pr98062.ps.gz ]. By using their approach, we derive the Riley–Mednykh polynomial for a family of [Formula: see text]-bridge knot orbifolds. As a result, we obtain explicit formulae for the volumes and Chern–Simons invariants of orbifolds and cone-manifolds on the knot with Conway’s notation [Formula: see text].

Diagram uniqueness for highly twisted plats

Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of Schubert who classified 2-bridge knots in the 1950s. We prove a uniqueness result for this enumeration. Using recent developments in geometric topology, including distances in the curve complex and techniques with incompressible surfaces, we show that infinitely many knot and link diagrams have a unique simple $m$-bridge diagram. Precisely, if $m$ is at least three, if each twist region of the diagram has at least three crossings, and if the length $n$ of the diagram is sufficiently long, i.e., $n>4m(m-2)$, then such a diagram is unique up to obvious rotations. This projection gives a canonical form for such knots and links, and thus provides a classification of these knots or links.

Classification of genus two knots which admit a (1,1)-decomposition

We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one (1,1)-knots, other consists of vertical bandings of two crosscap number two 2-bridge knots, and the last one consists of genus two, tunnel number one satellite knots.

Epimorphisms between 2–bridge knot groups and their crossing numbers

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.

Knot Concordance and Homology Sphere Groups

We study two homomorphisms to the rational homology sphere group. If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of the rational homology sphere group generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ is infinitely generated, and implies that a connected sum $K$ of 2-bridge knots is concordant to a knot with determinant 1 if and only if $K$ is smoothly slice. Furthermore, if $\beta$ denotes the homomorphism from the knot concordance group defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls.

On handlebody structures of rational balls

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an alternative definition of rational balls, $A_{m,n}$, bounding $L(p^2,pq-1)$ by their handlebody decompositions alone. We show that these two families coincide - answering a question of Kadokami and Yamada. To that end, we show that each $A_{m,n}$ admits a Stein filling of the standard contact structure, $\bar{\xi}_{st}$, on $L(p^2,pq-1)$ investigated by Lisca.

Bounds for minimum step number of knots confined to tubes in the simple cubic lattice

Knots are ubiquitous in nature and their analysis has important implications in a wide variety of fields including fluid dynamics, material science and molecular and structural biology. In many systems particles are found in crowded environments hence it is natural to rigorously characterize the properties of knots in confined volumes. In this work we combine analytical and numerical work on the simple cubic lattice to determine the minimal number of lattice steps, minimum step number, needed to make a knot inside a tubular region. Our complementary approaches help us establish a detailed enumeration of minimal knot lengths and/or conformations of knots in tubular regions. Analytical results characterize the types of knots and links that can be embedded in a tubular regions and determines the minimum number of steps required to construct all 2-bridge knots and links up to ten crossings in the -tube. Simulation results, on the other hand, estimate the minimum step number and provide exact trajectories of all knot types up to eight crossings for wider tubular regions. These findings not only determine what knots and links can be built in a highly confined volume but also provide further evidence that the minimum step number required to realize a knot type increases with confining volume.

The palette numbers of 2-bridge knots

We prove that for any odd [Formula: see text], the [Formula: see text]-palette number of any effectively [Formula: see text]-colorable [Formula: see text]-bridge knot is equal to [Formula: see text]. Namely, there is an effectively [Formula: see text]-colored diagram of the [Formula: see text]-bridge knot such that the number of distinct colors that appeared in the diagram is exactly equal to [Formula: see text].

The 𝔰𝔩3 colored Jones polynomials for 2-bridge links

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for [Formula: see text] and [Formula: see text] clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the [Formula: see text] and [Formula: see text] colored Jones polynomials of [Formula: see text]-bridge knots and links explicitly using twist formulas.

A family of bi-orderable non-fibered 2-bridge knot groups

We give a new infinite family of non-fibered 2-bridge knots whose knot groups are bi-orderable.

On the universal deformations for ${\rm SL}_2$-representations of knot groups

Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given SL_2-representation of a finitely generated group Pi over a field whose characteristic is not 2. We then show its connection with the character scheme for SL_2-representations of Pi when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

A note on the topological sliceness of some 2-bridge knots

AbstractWe use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson–Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson–Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.

A class of knots with simple SU(2)-representations

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an infinite family of knots $K$ with bridge number $\geq 3$ which are $SU(2)$-simple. One expects the instanton knot Floer homology $I^\natural(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det(K)$. In fact, the complex underlying $I^\natural(K)$ is of rank equal to $\det(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of $SU(2)$-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.

Riley's conjecture on SL(2,R) representations of 2-bridge knots

We prove a conjecture of Riley on [Formula: see text] representations of 2-bridge knot groups. As a consequence we show that if [Formula: see text] is a 2-bridge knot with non-zero signature, then the fundamental group of the [Formula: see text]-fold cyclic branched cover of [Formula: see text] is left-orderable for [Formula: see text] sufficiently large.

A note on applications of the d-invariant and Donaldson's theorem

This paper contains two remarks about the application of the [Formula: see text]-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750] concerning Conway mutation of alternating links.

Character varieties, A–polynomials and the AJ conjecture

We establish some facts about the behavior of the rational-geometric subvariety of the $SL_2(\c)$ or $PSL_2(\c)$ character variety of a hyperbolic knot manifold under the restriction map to the $SL_2(\c)$ or $PSL_2(\c)$ character variety of the boundary torus, and use the results to get some properties about the A-polynomials and to prove the AJ conjecture for certain class of knots in $S^3$ including in particular any $2$-bridge knot over which the double branched cover of $S^3$ is a lens space of prime order.

A note on Riley polynomials of 2-bridge knots

In this short note we show the existence of an epimorphism between groups of 2-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of 2-bridge knots by Riley polynomials.

A note on the OU sequences of a 2-bridge knot

An OU sequence is a cyclically ordered sequence in symbols [Formula: see text] and [Formula: see text] such that the number of [Formula: see text]’s is equal to that of [Formula: see text]’s. Every knot diagram defines an O’U sequence by reading [Formula: see text] and [Formula: see text] at over- and under-crossings, respectively, appeared along the diagram. In this note, we determine the OU sequences for a [Formula: see text]-bridge knot arising from its diagrams with two over-bridges.

Monodromy Maps of Fibered 2-Bridge Knots as Elements in Automorphism Groups of Free Groups

Monodromy Maps of Fibered 2-Bridge Knots as Elements in Automorphism Groups of Free Groups