Pretzel links, mutation, and the slice-ribbon conjecture

@article{Aceto2018PretzelLM,
  title={Pretzel links, mutation, and the slice-ribbon conjecture},
  author={Paolo Aceto and Min Hoon Kim and Junghwan Park and Arunima Ray},
  journal={Mathematical Research Letters},
  year={2018}
}
Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. Combining this result with previous work of the first author we prove the slice-ribbon conjecture for 4-stranded 2-component pretzel links. 
[PDF] Semantic Reader
3 Citations

Figures from this paper

3 Citations

Doubly Slice Montesinos Links

This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly…

The equivariant concordance group is not abelian

. We prove that the equivariant concordance group (cid:101) C is not abelian by exhibiting an infinite family of nontrivial commutators.

The equivariant concordance group is not abelian

References

SHOWING 1-10 OF 41 REFERENCES

The slice-ribbon conjecture for 3-stranded pretzel knots

We determine the (smooth) concordance order of the 3-stranded pretzel knots $P(p, q, r)$ with $p, q, r$ odd. We show that each one of finite order is, in fact, ribbon, thereby proving the…

Distinguishing mutant pretzel knots in concordance

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction…

On the slice-ribbon conjecture for pretzel knots

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots P.p1;:::;pn/ with one pi even. The 3‐stranded case yields two interesting families of…

On the slice-ribbon conjecture for Montesinos knots

We establish the slice-ribbon conjecture for a family P of Montesinos knots by means of Donaldson’s theorem on the intersection forms of definite 4-manifolds. The 4-manifolds that we consider are…

The topological sliceness of 3-strand pretzel knots

We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon…

The Effect of Mutation on Link Concordance, 3-manifolds and the Milnor Invariants

We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We show…

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can…

Mutation of knots

In general, mutation does not preserve the Alexander module or the concordance class of a knot. For a discussion of mutation of classical links, and the invariants which it is known to preserve, the…

Lens spaces, rational balls and the ribbon conjecture

We apply Donaldson's theorem on the intersection forms of definite 4-manifolds to characterize the lens spaces which smoothly bound rational homology 4-dimensional balls. Our result implies, in…

Slice Implies Mutant Ribbon for Odd, 5-Stranded Pretzel Knots

A pretzel knot $K$ is called $odd$ if all its twist parameters are odd, and $mutant$ $ribbon$ if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots…