Alternating links and definite surfaces

@article{Greene2017AlternatingLA,
  title={Alternating links and definite surfaces},
  author={Joshua Evan Greene},
  journal={Duke Mathematical Journal},
  year={2017},
  volume={166},
  pages={2133-2151}
}
  • J. Greene
  • Published 19 November 2015
  • Mathematics
  • Duke Mathematical Journal
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juh\'asz and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition. 

Figures from this paper

Geometry of alternating links on surfaces
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising theExpand
A Jones slopes characterization of adequate knots
We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can beExpand
A characterization of adequate knots
We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can beExpand
Jones slopes and coarse volume of near-alternating links
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their coloredExpand
On eigenvalues of double branched covers
  • Kouki Sato
  • Mathematics
  • Proceedings of the American Mathematical Society
  • 2019
For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, theExpand
Alternating links have at most polynomially many Seifert surfaces of fixed genus
Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial inExpand
A characterization of alternating link exteriors in terms of cubed complexes
  • Shunsuke Sakai
  • Mathematics
  • Journal of Knot Theory and Its Ramifications
  • 2018
We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a “signed BW cubed-complex”, and give a characterization for a signed BWExpand
Odd Order Group Actions on Alternating Knots
Let K be a an alternating prime knot in the 3-sphere. We investigate the category of flypes between reduced alternating diagrams for K. As a consequence, we show that any odd prime order action on KExpand
A topological characterization of toroidally alternating knots
  • Seungwon Kim
  • Mathematics
  • Communications in Analysis and Geometry
  • 2019
We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficientExpand
Quandle coloring conditions and zeros of the Alexander polynomials of Montesinos links
We give a simple condition for the existence of a nontrivial quandle coloring on a Montesinos link, which describes the distribution of the zeros of the Alexander polynomial. By this condition, weExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 51 REFERENCES
A characterisation of alternating knot exteriors
We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior andExpand
Minimal Genus Seifert Surfaces for Alternating Links
We give a complete proof of results announced by Hirasawa and Sakuma describing explicitly the Kakimizu complex of a non-split, prime, special, alternating link.
Closed incompressible surfaces in alternating knot and link complements
THEOREM 2 (The Meridian Lemma). Zf L is a non-split prime alternating link, and if S c S3 - L is a closed incompressible surface, then S contains a circle which is isotopic in S3 - L to a meridian ofExpand
A spanning tree expansion of the jones polynomial
A NEW combinatorial formulation of the Jones polynomial of a link is used to establish some basic properties of this polynomial. A striking consequence of these properties is the result that a linkExpand
Jones polynomials and classical conjectures in knot theory
The primeness is necessary in the last statement ofTheorem B, since the connected sum of two figure eight knots is alternating, but it has a minimal non-alternating projection. Note that the figureExpand
Signatures of links in rational homology spheres
Abstract A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant underExpand
Kauffman's polynomial and alternating links
These theorems result in quick tests which will often distinguish between links having alternating diagrams with the same number of crossings; links having reduced alternating diagrams with differentExpand
Lattices, graphs, and Conway mutation
The d-invariant of an integral, positive definite lattice Λ records the minimal norm of a characteristic covector in each equivalence class $({\textup{mod} \;}2\varLambda)$. We prove that theExpand
Discrete Torelli theorem
For an algebraic curve that has only simplest singularities and only rational irreducible components, the generalized Jacobian coincides with the moduli variety of topologically trivial linearExpand
Algorithmic Topology and Classification of 3-Manifolds
Simple and special polyhedra.- Complexity theory of 3-manifolds.- Haken theory of normal surfaces.- Applications of the theory of normal surfaces.- Algorithmic recognition of S3.- Classification ofExpand
...
1
2
3
4
5
...