Crowell’s Derived Group and Twisted Polynomials
نویسندگان
چکیده
The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell’s construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander modules and polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered. Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.
منابع مشابه
Representations of Knot Groups and Twisted Alexander Polynomials
We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.
متن کاملOn L2 –eigenfunctions of Twisted Laplacian on Curved Surfaces and Suggested Orthogonal Polynomials
We show in a unified manner that the factorization method describes completely the L2 -eigenspaces associated to the discrete part of the spectrum of the twisted Laplacian on constant curvature Riemann surfaces. Subclasses of two variable orthogonal polynomials are then derived and arise by successive derivations of elementary complex valued functions depending on the geometry of the surface. M...
متن کاملNote on q-Extensions of Euler Numbers and Polynomials of Higher Order
In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted (h, q)-extension of Euler polynomials and numbers, by using p-adic q-deformed fermionic integral on Zp. By applying their generating functions, they derived the complete sums of products of the twisted (h, q)-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new q-extension o...
متن کاملResearch Article Some Relationships between the Analogs of Euler Numbers and Polynomials
We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distrib...
متن کاملCovering morphisms of groupoids, derived modules and a 1-dimensional Relative Hurewicz Theorem∗
We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces, covering morphisms of groupoids, and Crowell’s notion of derived modules.
متن کامل