The guts of a knot is an invariant defined for the knot complement by Agol-Zhang. Nearly fibered knots, which are defined as knots whose Floer homology has dimension two in the top Alexander grading, were introduced by Baldwin-Sivek. We provide three models for the guts of nearly fibered knots in the 3-sphere. As a corollary, the nearly fibered condition can be purely topologically characterized and is independent of the specific version of Floer theory.

This paper studies the existence of 2-torsion in instanton Floer homology with Z coefficients for closed 3-manifolds and singular knots. First, we show that the non-existence of 2torsion in the framed instanton Floer homology Ia(Sn3 (K); Z) of any nonzero integral n-surgery along a knot Kin S3 would imply that K is fibered. Also, we show that Ia(Sr3(K);Z) for any nontrivial K with r = 1, 1/2, 1/4 always has 2-torsion. These two results indicate that the existence of 2-torsion is expected to be a generic phenomenon for Dehn surgeries along knots. Second, we show that for genus-one knots with nontrivial Alexander polynomials and for unknottingnumber-one knots, the unreduced singular instanton knot homology Ia(S3,K;Z) always has 2-torsion. Finally, some crucial lemmas that help us demonstrate the existence of 2torsion are motivated by analogous results in Heegaard Floer theory, which may be of independent interest. In particular, we show that, for a knot Kin S3, if there is a nonzero rational number r such that the dual knot Krr inside Sr3 (K) is Floer simple, then Sr3 (K) must be an L-space and K must be an L-space knot. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Suppose K ⊂ S^3 is a knot and suppose p and q are co-prime integers with q≥ 1. For any field 𝕂, we establish a dimension formula for the framed instanton homology of knot surgeries: I^♯(S^3_p/q(K); 𝕂) = q · r_𝕂(K) + |p - q · ν^♯_𝕂(K)| for certain integers r_𝕂(K) and ν^♯_𝕂(K), except possibly when p/q = ν^♯_𝕂(K) and ν^♯_𝕂(K) is even. This formula generalizes the result of Baldwin–Sivek from the case 𝕂 = ℂ to arbitrary fields. Based on the result for 𝕂 = ℤ/2, we obtain that S^3_p/q(K) is not SU(2)-abelian for any knot K other than the unknot and the right-handed trefoil whenever p/q ∈ [0,6) and p ∈{ a^e, 2a^e } for some prime number a and natural number e, thereby extending existing results for p/q ∈ [0,5] and p = a^e. A byproduct of the techniques developed in this paper is that we generalize the distance-two surgery exact triangle by Culler–Daemi–Xie and Daemi–Miller-Eismeier–Lidman from ℤ/2 coefficients to any coefficient ring.

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous knots in any 3-manifold, and a formula encoding a large portion of $I^\sharp(S^3_0(K))$. Second, we use the integral surgery formula to study the framed instanton homology of many 3-manifolds: Seifert fibered spaces with nonzero orbifold degrees, especially nontrivial circle bundles over any orientable surface, surgeries on a family of alternating knots and all twisted Whitehead doubles, and splicings with twist knots. Finally, we use the previous techniques and computations to study almost L-space knots, ${\it i.e.}$, the knots $K\subset S^3$ with $\dim I^\sharp(S_n^3(K))=n+2$ for some $n\in\mathbb{N}_+$. We show that an almost L-space knot of genus at least $2$ is fibered and strongly quasi-positive, and a genus-one almost L-space knot must be either the figure eight or the mirror of the $5_2$ knot in Rolfsen's knot table.

For any knot K in S^3 and any positive rational r, we show that smooth (-r)-surgery on K always admits a tight contact structure. More specifically, the tightness is detected by the non-vanishing Heegaard Floer contact invariant.

In our earlier work on 2-torsion in instanton Floer homology, we considered only integral surgeries on a knot K⊂ S^3 and showed that the absence of 2-torsion forces K to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology I^♯(S^3_r(K);ℤ) is 2-torsion-free for some r∈ℚ_+, then K is an instanton L-space knot and r>2g(K)-1. Leveraging this 2-torsion perspective, we also obtain new small-surgery obstructions: If either S^3_5(K) or S^3_11/2(K) is SU(2)-abelian, then K must be the unknot or the right-handed trefoil. This result sharpens the small-SU(2)-abelian surgery theorems of Kronheimer–Mrowka, Baldwin–Sivek, and Baldwin–Li–Sivek–Ye.

We prove that the unreduced singular instanton homology I^♯(Y,K;ℤ) has 2-torsion for any null-homologous fibered knot K of genus g>0 in a closed 3-manifold Y except for #^2gS^1× S^2. The main technical result is a formula of I^♯(Y,K;ℂ) via sutured instanton theory, by which we can compare the dimensions of I^♯(Y,K;𝔽_2) and I^♯(Y,K;ℂ). As a byproduct, we show that I^♯(S^3,K;ℂ) for a knot K⊂ S^3 admitting lens space surgeries is determined by the Alexander polynomial, while some special cases of torus knots have been previously studied by many people. Another byproduct is that the next-to-top Alexander grading summand of instanton knot homology KHI(S^3,K,g(K)-1) is non-vanishing when K has unknotting number one, which generalizes the Baldwin–Sivek's result in the fibered case. Finally, we discuss the relation to the Heegaard Floer theory.

We construct cobordism maps for the \textit{minus} version of instanton knot homology associated to a \textit{specially decorated} knot cobordisms of arbitrary genus between two null-homologous knots in closed oriented $3$-manifolds. As an application of our construction, we recover an inequality between the torsion order of knots in instanton theory, which was originally established in Heegaard Floer theory by work of Juh\'asz, Miller, and Zemke. We further use this inequality to compute the framed instanton Floer homology of any non-zero Dehn surgeries along an alternating knot of bridge index at most $3$.

We prove an integral surgery formula for framed instanton homology I (Ym(K)) for any knot K in a 3-manifold Y with [K] D 0 2 H1(YI Q) and m 0. Although the statement is similar to Ozsv & aacute;th-Szab & oacute;'s integral surgery formula for Heegaard Floer homology, the proof is new and based on sutured instanton homology SHI and the octahedral lemma in the derived category. As byproducts, we obtain a formula computing instanton knot homology of the dual knot analogous to Eftekhary's and Hedden-Levine's work, and also an exact triangle between I (Ym(K)), I (YmCk(K)) and k copies of I (Y ) for any m 0 and large k. In the proof of the formula, we discover many new exact triangles for sutured instanton homology and relate some surgery cobordism map to the sum of bypass maps, which are of independent interest. In a companion paper, we derive many applications and computations based on the integral surgery formula.

We define a 1-parameter family of homology invariants for links in thickened oriented surfaces. It recovers the homology invariant of Asaeda-Przytycki-Sikora (arxiv:0409414) and the invariant defined by Winkeler (arxiv:2106.03834). The new invariant can be regarded as a deformation of Asaeda-Przytycki-Sikora homology; it is not a Lee-type deformation as the deformation is only non-trivial when the surface is not simply connected. Our construction is motivated by computations in singular instanton Floer homology. We also prove a detection property for the new invariant, which is a stronger result than the main theorem of arxiv:2208.13963.

We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying , defined by the second author via the minus flavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots.

We show that in (S^3,ξ_std) if K is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. K has a Legendrian representative Λ with tb(Λ)-rot(Λ)=2g(K)-1), then K has a Legendrian representative L with tb=0. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in 3-manifolds other than S^3. We also show that if K is a nearly fibered knot in S^3 then τ(K)=g(K) implies that K realizes the three-dimensional Thurston-Bennequin bound.

We prove that the fundamental group of 3-surgery on a nontrivial knot in the 3-sphere always admits an irreducible SU(2)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the Property P conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery slopes in the interval [3,5).

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $\chi_{\rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $\chi_{\rm gr}(SHI(M,\gamma))=\chi_{\rm gr}(SFH(M,\gamma))$ for any balanced sutured manifold $(M,\gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $\chi_{\rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $\widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $\mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^\sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $\underline{\rm KHI}^-(Y,K)$ introduced by the first author.

In this paper, we generalize the work of the second author in Li (Direct systems and the knot monopole Floer homology, 2019. arXiv:1901.06679 ) and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handlebodies.. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász (Geom Topol 14(3):1303–1354, 2010) for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka (J Differ Geom 84(2):301–364, 2010). The same result was originally proved by Ozsváth and Szabó for link Floer homology in Ozsváth and Szabó (J Am Math Soc 21(3):671–709, 2008). Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author in Li (2019), to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in $$S^3$$ . This result was originally achieved by Kronheimer and Mrowka (2010).

Suppose $\mathcal {H}$ is an admissible Heegaard diagram for a balanced sutured manifold $(M,\gamma )$ . We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology $\mathit {SHI}(M,\gamma )$ . It follows, in particular, that strong L-spaces are instanton L-spaces.

Suppose Σ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of (-1,1)×Σ . We prove that the Asaeda–Przytycki–Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in {0}×Σ . As a consequence, the APS homology detects the unknot in (-1,1)×Σ . This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in Xie and Zhang (Instantons and Khovanov skein homology on I× T^2 , 2020. arXiv:2005.12863 ). Our proof is different from the previous detection results obtained by instanton homology because in this case, the second page of Kronheimer–Mrowka’s spectral sequence is not isomorphic to the APS homology. We also characterize all links in product manifolds that have minimal sutured instanton homology, which may be of independent interest.

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a $3$-manifold or a null-homologous knot inside a $3$-manifold and the Heegaard diagram of that $3$-manifold or knot. We further use this relation to compute the instanton knot homology of some families of $(1,1)$-knots, including all torus knots in $S^3$, which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology $KHI(Y,K)$ and the framed instanton Floer homology $I^\sharp(Y)$. In particular, we prove the inequality $\dim_\mathbb{C} I^\sharp(Y)\le \dim_\mathbb{C}KHI(Y,K)$ for all rationally null-homologous knots $K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along $K$ that corresponds to the decomposition along torsion spin$^c$ decompositions in monopole and Heegaard Floer theory.

This paper establishes a new technique that enables us to access some fundamental structural properties of instanton Floer homology. As an application, we establish, for the first time, a relation between the instanton Floer homology of a 3 $\hskip.001pt 3$ -manifold or a null-homologous knot inside a 3 $\hskip.001pt 3$ -manifold and the Heegaard diagram of that 3 $\hskip.001pt 3$ -manifold or knot. We further use this relation to compute the instanton knot homology of some families of ( 1 , 1 ) $(1,1)$ -knots, including all torus knots in S 3 $S^3$ , which were mostly unknown before. As a second application, we also study the relation between the instanton knot homology K H I ( Y , K ) $KHI(Y,K)$ and the framed instanton Floer homology I ♯ ( Y ) $I^\sharp (Y)$ . In particular, we prove the inequality dim C I ♯ ( Y ) ⩽ dim C K H I ( Y , K ) $\dim _\mathbb {C} I^\sharp (Y)\leqslant \dim _\mathbb {C}KHI(Y,K)$ for all rationally null-homologous knots K ⊂ Y $K\subset Y$ and we constructed a new decomposition of the framed instanton Floer homology of Dehn surgeries along K $K$ that corresponds to the decomposition along torsion spin c ${}^c$ decompositions in monopole and Heegaard Floer theory.

The guts of a knot is an invariant defined for the knot complement by Agol-Zhang. Nearly fibered knots are defined to be knots whose Floer homology has dimension two in the top Alexander grading and were introduced by Baldwin-Sivek. In this note, we identify three models that describe the guts of any nearly fibered knots in the 3-sphere. As a corollary, the nearly fibered condition can be purely characterized topologically and is independent of the specific version of Floer theory