Narrow Results

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation. We then sharpen our results and obtain various congruences guaranteeing nonuniqueness. We also show that a gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.

Let K be a two-bridge knot in S³. Then K is also denoted as the four-plat, b(p,q) to indicate its association with some rational number p/q.

The lens space L = L(p,q) admits an isometry, τ, of order 2, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K.

In this paper, the isometries of these orbifolds are classified; this is equivalent to computing the isometries of (S³,K).

The quantum SO(3)-invariants of ℚ-homology 3-spheres can be perturbatively expanded to the Ohtsuki invariant in number theory. On the other hand, it is known that the quantum SU(2)-invariants of 3-manifolds M admits a refinement involving a mod 2 cohomology class of M. A motivation of this paper is to study whether a perturbative expansion can be derived from the refinement. In this paper, we shall prove to be able to derive the perturbative expansion in the case of lens spaces by a concrete calculation, and shall define the perturbative invariant. Furthermore we obtain some properties for the perturbative invariant for covering spaces and connected sums of lens spaces.

Let K be a two-bridge knot or link in S³. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S³. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.

We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.

We give the explicit formula of the spin-refined Reshetikhin–Turaev SU(2) invariants of lens spaces. Using this result, we also give the formula of the spin-refined Turaev–Viro SU(2) invariants of lens spaces.

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p, q), that is the counterimage of L under the universal covering of L(p, q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p, q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift in S³. Using this construction, we are able to find different knots and links in L(p, q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.

In this paper, we calculate the values of the E₆ state sum invariant for the lens spaces L(p, q). In particular, we show that the values of the invariant are determined by p mod 12 and q mod (p, 12). As a corollary, we show that the E₆ state sum is a homotopy invariant for the oriented lens spaces.

In this paper, we work toward the Homflypt skein module of the lens spaces $L(p,1)$, ${𝒮}(L(p,1))$ using braids. In particular, we establish the connection between ${𝒮}(\mathrm{\text{ST}})$, the Homflypt skein module of the solid torus ST, and ${𝒮}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of ${𝒮}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, $\mathrm{\Lambda }$, of ${𝒮}(\mathrm{\text{ST}})$ presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014, arXiv:1412.3642 [math.GT]]. We show that ${𝒮}(L(p,1))$ is obtained from ${𝒮}(\mathrm{\text{ST}})$ by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis $\mathrm{\Lambda }$, where the bbm are performed on any moving strand of each element in $\mathrm{\Lambda }$. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set $\mathrm{\Lambda }$. The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. $3$-manifolds, since any $3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.

In this paper, the properties of the Kauffman bracket skein module (KBSM) of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the $3$-sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the $3$-sphere.

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, ${𝒮}(\mathrm{\text{ST}})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set ${\mathrm{\Lambda }}^{\mathrm{\text{aug}}}$, augmenting the basis $\mathrm{\Lambda }$ of ${𝒮}(\mathrm{\text{ST}})$.

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid monoids $T{M}_{g,n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an $L$-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

We compute the action of the ${𝔤𝔩}_2$-skein algebra of the torus on the ${𝔤𝔩}_2$-skein module of the solid torus. As a result, we show that the ${𝔤𝔩}_2$-skein modules of lens spaces is spanned by $(\lfloor \frac{p}{2}\rfloor +1)(2\lfloor \frac{p}{2}\rfloor +1)$ elements.

An explicit formula for the absolute value of the Witten invariants is derived. We discuss the relation between homotopy equivalence and the absolute value of Witten invariants for lens spaces. We also give examples of arbitrarily finitely many lens spaces which have the same Witten invariants for any level r.

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces.

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\ne 0$. 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, ${\mathrm{\text{TL}}}_{1,n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on ${\mathrm{\text{TL}}}_{1,n}$. The universal invariant $V$ is equivalent to the KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description of $L(p,q)$, obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM($L(p,q)$). We first present the solution for the case $q=1$, which corresponds to obtaining a new basis, ${{ℬ}}_p$, for KBSM($L(p,1)$) with $(\lfloor p/2\rfloor +1)$ elements. We note that the basis ${{ℬ}}_p$ is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case $q>1$, we first show how the new basis ${{ℬ}}_p$ of KBSM($L(p,1)$) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case $q>1$. 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.

Let be a spatial graph in the 3-sphere which covers a graph in the lens space. We introduce a combinatorial representation of which reflects the symmetry of the spatial graph. This representation involves a ribbon n-graph and the generator of the center of the n-braid group. We apply this result to study the Yamada polynomial of a class of lens spatial graphs.