Jonathan Fine
A complete $g$-vector for convex polytopes.
Tue, 01/12/2010 - 16:01 — SomNambulAuthors: Jonathan Fine
Subjects: Combinatorics
AbstractWe define an extension of the toric (middle perversity intersection homology)
$g$-vector of a convex polytope $X$. The extended $g(X)$ encodes the whole of
the flag vector $f(X)$ of $X$, and so is called complete. We find that for many
examples that $g_k(X)\geq 0$ for most $k$ (independent of $X$).A complete h-vector for convex polytopes.
Tue, 12/01/2009 - 16:01 — SomNambulAuthors: Jonathan Fine
Subjects: Combinatorics
AbstractThis note defines a complete h-vector for convex polytopes, which extends the
already known toric (or mpih) h-vector and has many similar properties.
Complete means that it encodes the whole of the flag vector.First we define the concept of a generalised h-vector and state some
properties that follow. The toric h-vector is given as an example. We then
define a complete generalised h-vector, and again state properties. Finally, we
show that this complete h-vector and all with similar properties will sometimes
have negative coefficients.A note on braids and Parseval's theorem.
Tue, 11/24/2009 - 16:01 — SomNambulAuthors: Jonathan Fine
Subjects: Quantum Algebra
AbstractIn 1988 Falk and Randell, based on Arnol'd's 1969 paper on braids, proved
that the pure braid groups are residually nilpotent. They also proved that the
quotients in the lower central series are free abelian groups.A filtration question on Belyi pairs and dessins.
Thu, 10/01/2009 - 00:35 — SomNambulAuthors: Jonathan Fine
Subjects: Algebraic Geometry
AbstractA Bely\u{\i} pair is a holomorphic map from a Riemann surface to $S^2$ with
additional properties. A dessin d'enfants is a bipartite graph with additional
structure. It is well know that there is a bijection between Bely\u{i} pairs
and dessins d'enfants.Vassiliev has defined a filtration on formal sums of isotopy classes of
knots. Motivated by this, we define a filtration on formal sums of Bely\u{\i}
pairs, and another on dessin d'enfants. We ask if the two definitions give the
same filtration.Parseval's theorem and Vassiliev-Kontsevich knot invariants.
Tue, 09/29/2009 - 15:21 — SomNambulAuthors: Jonathan Fine
Subjects: Quantum Algebra
AbstractThis paper use Parseval's theorem on Fourier series to solve the equation
$e^\tau = q$ for $\tau$ a Laurent series in $q$. It then states, as a
conjecture, an extension of this result to knots. The extension is that the
Vassiliev-Kontsevich invariants of a knot can be lifted to convergent sums of
knots in such a way that each knot is isotopic to the sum of its
Vassiliev-Kontsevich invariants. The proof of such a result seems to require a
Plancherel theorems for braid groupsParseval's theorem and Vassiliev-Kontsevich knot invariants.
Tue, 09/29/2009 - 15:21 — SomNambulAuthors: Jonathan Fine
Subjects: Quantum Algebra
AbstractThis paper use Parseval's theorem on Fourier series to solve the equation
$e^\tau = q$ for $\tau$ a Laurent series in $q$. It then states, as a
conjecture, an extension of this result to knots. The extension is that the
Vassiliev-Kontsevich invariants of a knot can be lifted to convergent sums of
knots in such a way that each knot is isotopic to the sum of its
Vassiliev-Kontsevich invariants. The proof of such a result seems to require a
Plancherel theorems for braid groups
User login
