Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userL-space knots are fibered and strongly quasipositive
255
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give a new, conceptually simpler proof of the fact that knots in $S^3$ with positive L-space surgeries are fibered and strongly quasipositive. Our motivation for doing so is that this new proof uses comparatively little Heegaard Floer-specific machinery and can thus be translated to other forms of Floer homology. We carried this out for instanton Floer homology in our recent article Instantons and L-space surgeries, and used it to generalize Kronheimer and Mrowkas results on $SU(2)$ representations of fundamental groups of Dehn surgeries.
rate research
Read More
Every $L$-space knot is fibered and strongly quasi-positive, but this does not hold for $L$-space links. In this paper, we use the so called H-function, which is a concordance link invariant, to introduce a subfamily of fibered strongly quasi-positive $L$-space links. Furthermore, we present an infinite family of $L$-space links which are not quasi-positive.
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenows in the (strong) positive. As a second main result, we give simple and complete characterizations of link diagrams with quasipositive canonical surface (the surface produced by Seiferts algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.
This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalisations of the Levine-Tristram signature.
In Dunfields catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in $S^3$, we determine that $22$ have tunnel number $2$ while the remaining all have tunnel number $1$. Notably, these $22$ manifolds contain $9$ asymmetric L-space knot complements. Furthermore, using SnapPy and KLO we find presentations of these $22$ knots as closures of positive braids that realize the Morton-Franks-Williams bound on braid index. The smallest of these has genus $12$ and braid index $4$.
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $pi_1(S^3setminus K) to G$ that send meridians of $K$ to $C$ is almost parsimoniously $mathsf{#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $mathsf{#P}$-complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
