ترغب بنشر مسار تعليمي؟ اضغط هنا

Identifying lens spaces in polynomial time

49   0   0.0 ( 0 )
 نشر من قبل Greg Kuperberg
 تاريخ النشر 2015
  مجال البحث فيزياء
والبحث باللغة English
 تأليف Greg Kuperberg




اسأل ChatGPT حول البحث

We show that if a closed, oriented 3-manifold M is promised to be homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute both n and k in polynomial time in the size of the triangulation of M. The tricky part is the parameter k. The idea of the algorithm is to calculate Reidemeister torsion using numerical analysis over the complex numbers, rather than working directly in a cyclotomic field.



قيم البحث

اقرأ أيضاً

A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M ( where M does not contain any embedded Klein bottles), then k is a (1,1) knot. Elements of the proof include handle addition and Dehn filling results/techniques of Jaco, Eudave-Munoz and Gordon as well as structure results of Schultens on the Heegaard splittings of graph manifolds.
The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group $G$ with relations. A valid subgroup $H$ of index $d$ in $G$ leads to a magic state $left|psirightrangle$ in $d$-d imensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite contextual geometry. In the present work, we choose $G$ as the fundamental group $pi_1(V)$ of an exotic $4$-manifold $V$, more precisely a small exotic (space-time) $R^4$ (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean $mathbb{R}^4$). Our selected example, due to to S. Akbulut and R.~E. Gompf, has two remarkable properties: (i) it shows the occurence of standard contextual geometries such as the Fano plane (at index $7$), Mermins pentagram (at index $10$), the two-qubit commutation picture $GQ(2,2)$ (at index $15$) as well as the combinatorial Grassmannian Gr$(2,8)$ (at index $28$) , (ii) it allows the interpretation of MICs measurements as arising from such exotic (space-time) $R^4$s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of quantum gravity.
95 - Louis H Kauffman 2020
The purpose of this paper is to give a new basis for examining the relationships of the Affine Index Polynomial and the Sawollek Polynomial. Blake Mellor has written a pioneering paper showing how the Affine Index Polynomial may be extracted from the Sawollek Polynomial. The Affine Index Polynomial is an elementary combinatorial invariant of virtual knots. The Sawollek polynomial is a relative of the classical Alexander polynomial and is defined in terms of a generalization of the Alexander module to virtual knots that derives from the so-called Alexander Biquandle. The present paper constructs the groundwork for a new approach to this relationship, and gives a concise proof of the basic Theorem of Mellor extracting the Affine Index Polynomial from the Sawollek Polynomial.
The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us to derive all the relevant informations on the topology of the different fibers that belong to the fibered face.
136 - Sheng Bai 2021
Conway-normalized Alexander polynomial of ribbon knots depend only on their ribbon diagrams. Here ribbon diagram means a ribbon spanning the ribbon knot marked with the information of singularities. We further give an algorithm to calculate Alexander polynomials of ribbon knots from their ribbon diagrams.
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