arXiv subjects

Peter M. Johnson

Publications and source records attributed to Peter M. Johnson.

6 recordsLinked to original sources

arXiv2013

A graphical calculus for tangles in surfaces

We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link and tangle theory can be expected to have a variety of applications, including ones involving 3-manifolds. It opens the way to new approaches for defining `facial' state-sum invariants that depend in part on assigning substates to faces of tangle diagrams.

math.GT↗

arXiv2013

A state sum invariant of tangles in surfaces

In this paper we define a new state sum based on the regions defined by tangles on a surface which is an oriented closed surface with a finite number of open holes drilled. From this state sum we obtain an invariant of regular isotopy for the tangles named $u$-invariant. The values of the $u$-invariant are in $\mathbb{Z}[u]$, where $u$ is a primitive fifth root of the unity. Various basic properties of the invariant are proved and discussed. It can be specialized to invariants of framed links in $\mathbb{R}^3$. Its categoric aspect is emphasized: composition of tangles in surfaces correspond to matrix multiplication. The values of the $u$-invariant conjugate under taking the mirror of the tangle.

math.GT↗

arXiv2011

A weighted graph problem from commutative algebra

We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials.

math.AC↗

arXiv2011

Inverses of monomial Cremona maps

We show that monomial Cremona maps of degree d on P^n can have inverses whose degree d' is quite large (for d > 2, d' = ((d-1)^n - 1)/(d-2) occurs), and that the full list of degrees d' does not always form an interval. An easy method for inverting the maps is presented.

math.AG↗

arXiv2010

Toric ideals of normalized graph algebras

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations between these generators is generated by a set of binomials, defined graphically. The spectra of the original algebra and of its integral closure turn out to be canonically homeomorphic.

math.AC↗

arXiv2007

Foundations for abstract forcing

The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.

math.LO↗