54,151 research outputs found
Knot concordance and Heegaard Floer homology invariants in branched covers
By studying the Heegaard Floer homology of the preimage of a knot K in S^3
inside its double branched cover, we develop simple obstructions to K having
finite order in the classical smooth concordance group. As an application, we
prove that all 2-bridge knots of crossing number at most 12 for which the
smooth concordance order was previously unknown have infinite smooth
concordance order.Comment: Expanded references; 25 pages, 5 figure
Jamming probabilities for a vacancy in the dimer model
Following the recent proposal made by Bouttier et al [Phys. Rev. E 76, 041140
(2007)], we study analytically the mobility properties of a single vacancy in
the close-packed dimer model on the square lattice. Using the spanning web
representation, we find determinantal expressions for various observable
quantities. In the limiting case of large lattices, they can be reduced to the
calculation of Toeplitz determinants and minors thereof. The probability for
the vacancy to be strictly jammed and other diffusion characteristics are
computed exactly.Comment: 19 pages, 6 figure
Folding Polyominoes into (Poly)Cubes
We study the problem of folding a polyomino into a polycube , allowing
faces of to be covered multiple times. First, we define a variety of
folding models according to whether the folds (a) must be along grid lines of
or can divide squares in half (diagonally and/or orthogonally), (b) must be
mountain or can be both mountain and valley, (c) can remain flat (forming an
angle of ), and (d) must lie on just the polycube surface or can
have interior faces as well. Second, we give all the inclusion relations among
all models that fold on the grid lines of . Third, we characterize all
polyominoes that can fold into a unit cube, in some models. Fourth, we give a
linear-time dynamic programming algorithm to fold a tree-shaped polyomino into
a constant-size polycube, in some models. Finally, we consider the triangular
version of the problem, characterizing which polyiamonds fold into a regular
tetrahedron.Comment: 30 pages, 19 figures, full version of extended abstract that appeared
in CCCG 2015. (Change over previous version: Fixed a missing reference.
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . By a result of Thiagarajan
(1996 and 2002), these unfoldings are exactly the trace regular event
structures. Thiagarajan (1996 and 2002) conjectured that regular event
structures correspond exactly to trace regular event structures. In a recent
paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on
the striking bijection between domains of event structures, median graphs, and
CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we
proved that Thiagarajan's conjecture is true for regular event structures whose
domains are principal filters of universal covers of (virtually) finite special
cube complexes.
In the current paper, we prove the converse: to any finite 1-safe Petri net
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
This establishes a bijection between 1-safe Petri nets and finite special cube
complexes and provides a combinatorial characterization of trace regular event
structures.
Using this bijection and techniques from graph theory and geometry (MSO
theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet
another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that
the monadic second order logic of a 1-safe Petri net is decidable if and only
if its unfolding is grid-free.
Our counterexample is the trace regular event structure
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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