Narrow Results

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $𝔟(p,q)$ is minimal where $q\le 6$ or $p\le 100$.

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the augmented Dehn presentation of the knot group of a tame, prime, alternating knot in regular position, with a common projection has a finite and complete rewriting system. This provides an algorithm for solving the word problem with this presentation and we find an algorithm for solving the word problem with the Dehn presentation also.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

We propose a conjecture concerning 3-manifolds (which implies the ℚ conjecture of Myers') and construct a countable sequence of examples which supports it.

Let G be a finitely generated group, and let λ ∈ G. If there exists a knot k such that πk = π₁(S³\k) can be mapped onto G sending the longitude to λ, then there exists infinitely many distinct prime knots with the property. Consequently, if πk is the group of any knot (possibly composite), then there exists an infinite number of prime knots k₁, k₂, … and epimorphisms ⋯ → πk₂ → πk₁ → πk each perserving peripheral structures. Properties of a related partial order on knots are discussed.

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T²-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S²-knot groups, S²-knot groups, T²-knot groups, and S³-knot groups.

We give explicit palindrome presentations of the groups of rational knots, i.e. presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2, ℂ) character variety of twist knots.

We use curvature techniques from geometric group theory to produce examples of virtual knot groups that are not classical knot groups.

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. Unlike most other proofs of this result, this proof uses only classic results in topology and combinatorial group theory.

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s₀ such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s₀]) ⊂SL(2, ℂ). Let be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), and , where , μ ∈ ℤ[s₀]. The number μ can be recursively evaluated.

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as E³ (Euclidean 3-space), H³ (hyperbolic 3-space) and E^{2, 1} (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which the generators are send to conjugate elements, by analyzing the points of an algebraic variety, that we call the variety of affine c-representations ofG. Each point in this variety corresponds to a representation in the unit group of a quaternion algebra and their affine deformations.

The number of representations of a knot group is an invariant of knots. In this paper, we calculate these numbers associated to SL(2;ℤ/pℤ)-representations for all the knots in Rolfsen's knot table. Moreover, we show some properties of these numbers.

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups.

The complete classification of representations of the Trefoil knot group G in S³ and SL(2, ℝ), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the noncyclic groups of affine Euclidean isometries generated by two isometries μ and ν such that μ² = ν³ = 1, in particular those which are crystallographic. We also prove that there are no affine crystallographic groups in the three-dimensional Minkowski space which are quotients of G.

Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots satisfying wb(K) ≤ vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K) = br(K), the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group G_K, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group Ḡ_K, and we apply these methods to address Questions 6.1–6.3 and 6.5 raised by Hirasawa, Kamada and Kamada in their paper [Bridge presentation of virtual knots, J. Knot Theory Ramifications 20(6) (2011) 881–893].

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

Let $M$ be a non-abelian semi-direct product of a cyclic group $\mathbb{Z}/n$ and an elementary abelian $p$-group $A=\oplus k(\mathbb{Z}/p)$ of order $p^k$, $p$ being a prime and $gcd(n,p)=1$. Suppose that the knot group $G(K)$ of a knot $K$ in the $3$-sphere is represented on $M$. Then we conjectured (and later proved) that the twisted Alexander polynomial ${\mathrm{\Delta }}_{\gamma ,K}(t)$ associated to $\gamma :G(K)\to M\to GL(p^k,\mathbb{Z})$ is of the form: $\frac{{\mathrm{\Delta }}_K(t)}{1-t}F(t^n)$, where ${\mathrm{\Delta }}_K(t)$ is the Alexander polynomial of $K$ and $F(t^n)$ is an integer polynomial in $t^n$. In this paper, we present a proof of the following. For a $2$-bridge knot $K(r)$ in $H(p)$, if $n=2$ and $k=1$, then $F(t^2)$ is written as $f(t)f(t^{-1})$, where $H(p)$ is the set of $2$-bridge knots whose knot groups map on that of $K(1/p)$ with $p$ odd.

For a cyclic covering map $(\mathrm{\Sigma },K)\to ({\mathrm{\Sigma }}^{\prime },K^{\prime })$ between two pairs of a 3-manifold and a knot each, we describe the fundamental group ${\pi }_1(\mathrm{\Sigma }\setminus K)$ in terms of ${\pi }_1({\mathrm{\Sigma }}^{\prime }\setminus K^{\prime })$. As a consequence, we give an alternative proof for the fact that certain knots in $S^3$ cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group $G$ generated by the commutators and the $p$th power of each element of $G$ plays a key role.

A $2$-knot is a surface in $R^4$ that is homeomorphic to $S^2$, the standard sphere in $3$-space. A ribbon $2$-knot is a $2$-knot obtained from $m$$2$-spheres in $R^4$ by connecting them with $m-1$ pipes. Let $K^2$ be a ribbon 2-knot. The ribbon crossing number, denoted by $r$-$cr(K^2)$, is a numerical invariant of the ribbon $2$-knot $K^2$. In [T. Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications10 (2001) 999–1003] we showed that there exist just $17$ ribbon $2$-knots of the ribbon crossing number up to three. In this paper, we show that there exist no more than $111$ ribbon $2$-knots of ribbon crossing number four.

Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of $3$-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how $3$-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of $3$-dimensional surgery as rotations of the decompactified $2$-sphere. Each rotation produces a different decomposition of the $3$-sphere which corresponds to a different visualization of the $4$-dimensional process of $3$-dimensional surgery.