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1. CJM 2010 (vol 62 pp. 1058)
On a Conjecture of S. Stahl
S. Stahl conjectured that the zeros of genus polynomials are real. In
this note, we disprove this conjecture.
Keywords:genus polynomial, zeros, real Category:05C10 |
2. CJM 2009 (vol 61 pp. 1300)
Monodromy Groups and Self-Invariance For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is self-invariant with respect to $d$, or
$d$-invariant. We develop algebraic tools for investigating these
operations on polytopes, and in particular give a criterion on the
existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application,
we analyze properties of self-dual edge-transitive polyhedra and
polyhedra with two flag-orbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a self-dual
equivelar polyhedron which contains no polarity (duality of order
2). We also extend the concept of Petrie dual to higher dimensions,
and we show how it can be dealt with using self-invariance.
Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedra Categories:51M20, 05C25, 05C10, 05C30, 52B70 |
3. CJM 2008 (vol 60 pp. 958)
A Note on a Conjecture of S. Stahl S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640)
conjectured that the zeros of genus polynomial are real.
L. Liu and Y. Wang disproved this conjecture on the basis
of Example 6.7. In this note, it is pointed out
that there is an error in this example and a new generating matrix
and initial vector are provided.
Keywords:genus polynomial, zeros, real Categories:05C10, 05A15, 30C15, 26C10 |
4. CJM 2008 (vol 60 pp. 960)
5. CJM 1997 (vol 49 pp. 617)
On the zeros of some genus polynomials In the genus polynomial of the graph $G$, the coefficient of $x^k$
is the number of distinct embeddings of the graph $G$ on the
oriented surface of genus $k$. It is shown that for several
infinite families of graphs all the zeros of the genus polynomial
are real and negative. This implies that their coefficients, which
constitute the genus distribution of the graph, are log concave and
therefore also unimodal. The geometric distribution of the zeros
of some of these polynomials is also investigated and some new
genus polynomials are presented.
Categories:05C10, 05A15, 30C15, 26C10 |
6. CJM 1997 (vol 49 pp. 193)
Classifying PL $5$-manifolds by regular genus: the boundary case In the present paper, we face the problem of classifying classes of
orientable PL $5$-manifolds $M^5$ with $h \geq 1$ boundary components,
by making use of a combinatorial invariant called {\it regular genus}
${\cal G}(M^5)$. In particular, a complete classification up to
regular genus five is obtained:
$${\cal G}(M^5) = \gG \leq 5 \Longrightarrow M^5 \cong \#_{\varrho
- \gbG}(\bdo) \# \smo_{\gbG},$$
where $\gbG = {\cal G}(\partial M^5)$ denotes the regular genus of
the boundary $\partial M^5$ and $\smo_{\gbG}$ denotes the connected
sum of $h\geq 1$ orientable $5$-dimensional handlebodies
$\cmo_{\alpha_i}$ of genus $\alpha_i\geq 0$
($i=1,\ldots, h$), so that $\sum_{i=1}^h \alpha_i = \gbG.$
\par
Moreover, we give the following characterizations of orientable PL
$5$-manifolds $M^5$ with boundary satisfying particular conditions
related to the ``gap'' between ${\cal G}(M^5)$ and either
${\cal G}(\partial M^5)$ or the rank of their fundamental group
$\rk\bigl(\pi_1(M^5)\bigr)$:
$$\displaylines{{\cal G}(\partial M^5)= {\cal G}(M^5)
= \varrho \Longleftrightarrow M^5 \cong \smo_{\gG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-1 \Longleftrightarrow
M^5 \cong (\bdo) \# \smo_{\gbG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)-2 \Longleftrightarrow
M^5 \cong \#_2 (\bdo) \# \smo_{\gbG}\cr
{\cal G}(M^5) = \rk\bigl(\pi_1(M^5)\bigr)= \varrho \Longleftrightarrow
M^5 \cong \#_{\gG - \gbG}(\bdo) \# \smo_{\gbG}.\cr}$$
\par
Further, the paper explains how the above results (together with
other known properties of regular genus of PL manifolds) may lead
to a combinatorial approach to $3$-dimensional Poincar\'e Conjecture.
Categories:57N15, 57Q15, 05C10 |