Generalized Fibonacci groupsH(r,n,s) that are connected labelled oriented graph groups

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).

We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern–Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern–Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern–Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S3 colored by fundamental representation, the index equals the Poincare polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S3 has a large-N dual which is the refined topological string on \({X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}\) ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SLN knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S3.

A slope [Formula: see text] is called a characterizing slope for a given knot [Formula: see text] if whenever the [Formula: see text]-surgery on a knot [Formula: see text] is homeomorphic to the [Formula: see text]-surgery on [Formula: see text] via an orientation preserving homeomorphism, then [Formula: see text]. In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot [Formula: see text]. More explicitly besides all negative integer slopes there are [Formula: see text] slopes which were unknown to be characterizing for [Formula: see text], including [Formula: see text] nontrivial [Formula: see text]-space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope [Formula: see text] is a characterizing slope for [Formula: see text] if [Formula: see text] and [Formula: see text]. In particular every nontrivial [Formula: see text]-space slope of [Formula: see text] is characterizing for [Formula: see text]. More explicitly this work yields [Formula: see text] new characterizing slopes for [Formula: see text]. Another interesting consequence of this work is that if a nontrivial [Formula: see text]-surgery on a non-torus knot in [Formula: see text] yields a manifold of finite fundamental group, then [Formula: see text].

Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. We study the relationship between $SU(2)$ and $\sldos$-representations of this group, looking at their characters. Using the description of the character variety of $G$, $X(G)$, we give a geometric description of $Y(G)\subset X(G)$, the set of characters arising from $SU(2)$-representations.

Motive of the SL4-character variety of torus knots

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot $K$ admits a toroidal Seifert fibered surgery, then $K$ is either the trefoil knot and the surgery slope is zero, or the connected sum of a $(2,p)$-torus knot and a $(2,q)$-torus knot and the surgery slope is $2(p+q)$ with $|p|, |q| \\ge 3$.

We show that for almost all curves in given by an equation of the form , where and are coprime integers, the fundamental group of the complement of the curve has presentation , that is, it coincides with the group of the torus knot . In the projective case, for almost every curve in which is the projective closure of a curve in given by an equation of the form , the fundamental group of the complement is a free product with amalgamated subgroup of two cyclic groups of finite order. In particular, for the general curve given by the equation , where is a homogenous polynomial of degree , we have .

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

Given a knot [Formula: see text], we may construct a group [Formula: see text] from the fundamental group of [Formula: see text] by adjoining an [Formula: see text]th root of the meridian that commutes with the corresponding longitude. For [Formula: see text] these “generalized knot groups” determine [Formula: see text] up to reflection. The second author has shown that for [Formula: see text], the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues [Formula: see text], [Formula: see text], constructed as connected sums of [Formula: see text]-torus knots of opposite or identical chiralities. More precisely, for coprime [Formula: see text] and [Formula: see text] satisfying a coprimality condition with [Formula: see text] and [Formula: see text], we construct an explicit finite group [Formula: see text] (depending on [Formula: see text], [Formula: see text] and [Formula: see text]) such that [Formula: see text] and [Formula: see text] can be distinguished by counting homomorphisms into [Formula: see text]. The coprimality condition includes all [Formula: see text] coprime to [Formula: see text]. The result shows that the difference between these two groups can be detected using a finite group.

In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the reader's convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].

The twisted Reidemeister torsion of an iterated torus knot

Epimorphisms of knot groups onto free products

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link