arXiv research

Study calculates instanton Floer homology for surgeries on L-space knots.

New proof shows certain 3D shapes can't be instanton L-spaces.

Computes upper bounds for instanton knot homology.

New invariants from framed instanton homology for knot concordance.

Enhances knot surgery formulae for instanton Floer homology.

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…

The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.

New insights into knot surgeries via instanton 2-torsion.

Proves exact triangle linking knot instanton Floer homology to surgeries.

This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.

Enhanced Euler characteristic improves knot homology detection.

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the $\textrm{Spin}^c$ decompositions of cobo…

Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, whic…

(1,1) non-L-space knots are foliar in 3D space.

New infinite family of knots found with unique properties.

Paper constructs infinitely many non-braid positive hyperbolic L-space knots.

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifer…

Study shows 3D shapes can't fit in certain 4D spaces.

Study concordance of alternating torus knots to L-space knots.

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number $\geq 3$ are not $SU(2)$-simple. We provide an …

Paper classifies $CFK^\infty$ type of almost L-space knots.

The study examines quasi-alternating surgeries on knots and their properties.

New hyperbolic knots not concordant to algebraic ones found.

New infinite family of hyperbolic L-space knots with specific semigroups.

Two knots with unique surgery properties.

L-space knots lack essential Conway spheres, proven with Floer theory.

Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight …

We find braid positive presentations for most L-space knots, except one, and explore related knot properties.

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient…

New knots share same Upsilon invariant despite different Alexander polynomials.

Study on pretzel knots showing cyclic branched covers are L-spaces.

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.

New description of $(1,1)$ L-space knots leads to non-left-orderable surgeries.

The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.

Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots …

Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations…

In this paper we look at the knot complement problem for L-space $\mathbb{Z}$-homology spheres. We show that an L-space $\mathbb{Z}$-homology sphere $Y$ cannot be obtained as a non-trivial surgery along a knot $K\subset Y$. As a consequence, we prove that knots in an L-space $\mathbb{Z}$-homology sphere are determined …

For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves…

We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two in…

Formula for τ-invariant of satellite knots derived from L-space satellite operators.

We show there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot $T(2,2g+1)$ of the same genus and they are fibred and strongly quasipositive.

The study confirms conjectures about slopes of knots using knot Floer homology.

If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…

We give a formula of the Upsilon invariant of any L-space cable knot $K_{p,q}$ using $p,Υ_K$ and $Υ_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We compute the integral values for L-space iterated cable knots.