On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

Cutting Arcs For Torus Links And Trees

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., An , Dn , E6 , E7 and E8 ) among all positive tree-like Hopf plumbings.

Geometry of the SL(3,ℂ)-character variety of torus knots.

Let G be the fundamental group of the complement of the torus knot of type (m, n). It has a presentation G = . We find a geometric description of the character variety X(G) of characters of representations of G into SL(3,ℂ), GL(3,ℂ) and PGL(3,ℂ).

Copy of Gov[erno]r Philips's additional instructions about settling some Palatine families in Nova Scotia: Copy Col. Dunbar's additional instruction about settling some Irish families in Nova Scotia : manuscript, [1730]

Copies of two documents, approved 20 Apr. 1730; instructions to Richard Philips state that a number of Protestant Irish and Palatine families have been granted land in Nova Scotia, to be surveyed by David Dunbar; and instructions to Dunbar to survey and lay out land for the families.

Per Carlo Maria Maggi : inaugurandosi il monumento alle Scuole Palatine, nel IIo centenario della morte, 1630-1699 /

Copy 2: University of Illinois bookplate: "From the library of Conte Antonio Cavagna Sangiuliani di Gualdana Lazelada di Bereguardo, purchased 1921".

A royal cavalier : the romance of Rupert, prince palatine /

Mode of access: Internet.

Remains, historical and literary, connected with the Palatine counties of Lancaster and Chester.

Title varies slightly.

Early eighteenth century Palatine emigration; a British government redemptioner project to manufacture naval stores,

Bibliography: p. [229]-241; bibliographical foot-notes.

Mattice family history, 1709-1961; Palatine emigration.

Mode of access: Internet.

East Cheshire: past and present; or, A history of the hundred of Macclesfield, in the county palatine of Chester. From original records.

Mode of access: Internet.