51. CMB 2001 (vol 44 pp. 440)
 Hironaka, Eriko

The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the wellknown Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

52. CMB 2000 (vol 43 pp. 268)
 Bogley, W. A.; Gilbert, N. D.; Howie, James

Cockcroft Properties of Thompson's Group
In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of selfhomeomorphisms of the Cantor
set and showed, among other things, that $F$ is finitely presented.
Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed
that $F$ is the fundamental group of a finite twocomplex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the twosphere into $Z^2$ is
homologically trivial. We show that no proper covering complex of
$Z^2$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
twocomplex with fundamental group $F$ is Cockcroft.
Keywords:twocomplex, covering space, Cockcroft twocomplex, Thompson's group Categories:57M20, 20F38, 57M10, 20F34 

53. CMB 2000 (vol 43 pp. 343)
54. CMB 2000 (vol 43 pp. 145)
 Chang, JaeHo; Lee, Sang Youl; Park, ChanYoung

On the 2Parallel Versions of Links
In this paper, we show that the absolute value of the signature of
the $2$parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2parallel version Category:57M25 

55. CMB 2000 (vol 43 pp. 79)
56. CMB 1999 (vol 42 pp. 257)
57. CMB 1999 (vol 42 pp. 190)
 Gilmer, Patrick M.

Topological Quantum Field Theory and Strong Shift Equivalence
Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The TuraevViro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the SilverWilliams
invariant, by using the TQFT associated to $G$ in its unmodified form.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence Categories:57R99, 57M99, 54H20 

58. CMB 1999 (vol 42 pp. 248)
 Weber, Christian

The Classification of $\Pin_4$Bundles over a $4$Complex
In this paper we show that the Liegroup $\Pin_4$ is isomorphic to
the semidirect product $(\SU_2\times \SU_2)\timesv \Z/2$ where
$\Z/2$ operates by flipping the factors. Using this structure
theorem we prove a classification theorem for $\Pin_4$bundles over
a finite $4$complex $X$.
Categories:55N25, 55R10, 57S15 

59. CMB 1999 (vol 42 pp. 149)
 Boyer, S.; Zhang, X.

A Note on Finite Dehn Fillings
Let $M$ be a compact, connected, orientable 3manifold whose
boundary is a torus and whose interior admits a complete hyperbolic
metric of finite volume. In this paper we show that if the minimal
CullerShalen norm of a nonzero class in $H_1(\partial M)$ is
larger than $8$, then the finite surgery conjecture holds for $M$.
This means that there are at most $5$ Dehn fillings of $M$ which
can yield manifolds having cyclic or finite fundamental groups and
the distance between any slopes yielding such manifolds is at most
$3$.
Categories:57M25, 57R65 

60. CMB 1999 (vol 42 pp. 46)
 Dijkstra, Jan J.

Generic Partial TwoPoint Sets Are Extendable
It is shown that under $\ZFC$ almost all planar compacta that meet
every line in at most two points are subsets of sets that meet every
line in exactly two points. This result was previously obtained by the
author jointly with K.~Kunen and J.~van~Mill under the assumption that
Martin's Axiom is valid.
Category:57N05 

61. CMB 1999 (vol 42 pp. 52)
 Edmonds, Allan L.

Embedding Coverings in Bundles
If $V\to X$ is a vector bundle of fiber dimension $k$ and $Y\to X$
is a finite sheeted covering map of degree $d$, the implications
for the Euler class $e(V)$ in $H^k(X)$ of $V$ implied by the
existence of an embedding $Y\to V$ lifting the covering map are
explored. In particular it is proved that $dd'e(V)=0$ where $d'$
is a certain divisor of $d1$, and often $d'=1$.
Categories:57M10, 55R25, 55S40, 57N35 

62. CMB 1998 (vol 41 pp. 374)
 Young, Carmen M.

Normal invariants of lens spaces
We show that normal and stable normal invariants of polarized
homotopy equivalences of lens spaces $M = L(2^m;\r)$ and
$N = L(2^m;\s)$ are determined by certain $\ell$polynomials
evaluated on the elementary symmetric functions
$\sigma_i(\rsquare)$ and $\sigma_i(\ssquare)$. Each polynomial
$\ell_k$ appears as the homogeneous part of degree $k$ in the
Hirzebruch multiplicative $L$sequence. When $n = 8$, the
elementary symmetric functions alone determine the relevant normal
invariants.
Categories:57R65, 57S25 

63. CMB 1998 (vol 41 pp. 252)
64. CMB 1998 (vol 41 pp. 140)
65. CMB 1997 (vol 40 pp. 309)
 Hillman, J. A.; Sakuma, M.

On the homology of finite abelian coverings of links
Let $A$ be a finite abelian group and $M$ be a
branched cover of an homology $3$sphere, branched over a link $L$,
with covering group $A$. We show that $H_1(M;Z[1/A])$ is determined
as a $Z[1/A][A]$module by the Alexander ideals of $L$ and certain
ideal class invariants.
Keywords:Alexander ideal, branched covering, Dedekind domain,, knot, link. Category:57M25 

66. CMB 1997 (vol 40 pp. 370)
 Rolfsen, Dale; Zhongmou, Li

Which $3$manifolds embed in $\Triod \times I \times I$?
We classify the compact $3$manifolds whose boundary is a union of
$2$spheres, and which embed in $T \times I \times I$, where $T$ is a
triod and $I$ the unit interval. This class is described explicitly as
the set of punctured handlebodies. We also show that any $3$manifold
in $T \times I \times I$ embeds in a punctured handlebody.
Categories:57N10, 57N35, 57Q35 

67. CMB 1997 (vol 40 pp. 204)