ترغب بنشر مسار تعليمي؟ اضغط هنا

Noncyclic covers of knot complements

110   0   0.0 ( 0 )
 نشر من قبل Nathan Broaddus
 تاريخ النشر 2004
  مجال البحث
والبحث باللغة English
 تأليف Nathan Broaddus




اسأل ChatGPT حول البحث

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $Phi (c)$, such that if $K$ is a nontrivial knot in the three-sphere with a diagram with $c$ crossings and a particularly simple JSJ decomposition then the complement of $K$ has a finite-sheeted, noncyclic cover with at most $Phi (c)$ sheets.

قيم البحث

اقرأ أيضاً

251 - Eaman Eftekhary 2015
We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_isubset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $widehat{mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $widehat{mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$big|(h_infty^1-h_1^1)(h_infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)big|.$$ We also show that if splicing the complement of a knot $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
The physical 3d $mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at th e roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $hat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $hat{Z}_a(q)$ for some hyperbolic 3-manifolds.
115 - Sunghyuk Park 2020
The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities for positive braid knots.
Given a link in a 3-manifold such that the complement is hyperbolic, we provide two modifications to the link, called the chain move and the switch move, that preserve hyperbolicity of the complement, with only a relatively small number of manifold-l ink pair exceptions, which are also classified. These modifications provide a substantial increase in the number of known hyperbolic links in the 3-sphere and other 3-manifolds.
By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to s tudy knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us to characterise invertibility and other properties in the hyperbolic case. Moreover, with our construction we are able to detect both the trivial knotoid in the 2-sphere and the trivial planar knotoid.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا