In the spirit of Bar-Natan's formulation of Khovanov homology for tangles, we extend the framework of involutive Khovanov homology to involutive tangles. This enables a divide-and-conquer computation of involutive Khovanov homology and the equivariant Rasmussen invariant, which results in a significant speedup for the algorithmic computation. With this, we obtain new examples of strongly invertible knots for which no slice disk is smoothly isotopic rel boundary to its symmetric counterpart. In particular, we show that the Whitehead doubles of the pretzel knots P(-3, 3, -3) and P(-5, 5, -5) admit exotic pairs of slice disks.

We study symmetries in equivariant versions of Khovanov homology, which include (i) the construction of an involution sigma b for the U(2)-equivariant theory, (ii) an integral lifting nu b of the Shumakovitch operation nu, and (iii) splitting of the U specialIntscript-and U specialIntscript & times; U(1)-equivariant theories generalizing earlier work over F 2 . Finally, we relate these structures to the Rasmussen s-invariant over an arbitrary field F.

We show that Iida-Taniguchi's -valued slice-torus invariant cannot be realized as a linear combination of Rasmussen's -invariant, Ozsv & aacute;th-Szab & oacute;'s -invariant, the -concordance invariants (), Baldwin-Sivek's instanton -invariant, Daemi-Imori-Sato-Scaduto-Taniguchi's instanton -invariant and Sano-Sato's Rasmussen-type invariants .

Motivated by the y-ification of HOMFLY–PT homology by Gorsky and Hogancamp, and the 𝔰𝔩_2-action of Gorsky, Hogancamp, and Mellit, we construct y-ifications of Khovanov homology and its equivariant versions within Bar-Natan's framework for tangles, and define an action of the element e in 𝔰𝔩_2 on these y-ifications. We then prove that our construction is compatible with the previous ones under Rasmussen's spectral sequence from HOMFLY–PT homology to Khovanov homology. Our construction is elementary and well suited to diagrammatic manipulations and algorithmic implementations. As a result, we verify directly that these additional structures distinguish pairs of knots with identical Khovanov homology and HOMFLY–PT homology, in particular the Conway knot and the Kinoshita–Terasaka knot.

Abstract We show that Iida–Taniguchi's ‐valued slice‐torus invariant cannot be realized as a linear combination of Rasmussen's ‐invariant, Ozsváth–Szabó's ‐invariant, the ‐concordance invariants (), Baldwin–Sivek's instanton ‐invariant, Daemi–Imori–Sato–Scaduto–Taniguchi's instanton ‐invariant and Sano–Sato's Rasmussen‐type invariants .

For strongly invertible knots, we define an involutive version of Khovanov homology, and from it derive a pair of integer-valued invariants (s, sx), which is an equivariant version of Rasmussen's s-invariant. Using these invariants, we reprove that the infinite family of knots Jn introduced by Hayden each admits exotic pairs of slice disks. Our construction is intended to give a Khovanov-theoretic analogue of the formalism given by Dai, Mallick and Stoffregen in involutive knot Floer theory.

We determine the Khovanov-Rozansky HOMFLY homology for all prime knots with up to 11 crossings, by direct computations using an algorithm described in this paper. In addition, we determine their S-invariants, thereby extending the results of Chandler and Gorsky, which were based on a smaller dataset from an earlier version of this work.

We introduce a diagrammatic approach to Rasmussen's s-invariant, based on Bar-Natan's reformulation of Khovanov homology for tangles and cobordisms. This method enables a local computation of s from a tangle decomposition of a knot diagram. As an application, we compute the s-invariants of all 3-strand pretzel knots.

This paper is a continuation of our previous work, where we defined an embedded cobordism map on the instanton cube complex that recovers the cobordism maps both in Khovanov homology and singular instanton theory. In this paper, we extend this construction to immersed cobordisms. As an application, we show that, for any smooth, oriented (not necessarily ribbon) concordance C from a two-bridge torus knot, the induced map Kh(C) on reduced Khovanov homology is injective, with the left inverse given by the reversal of C.

We show that Iida–Taniguchi's ℤ-valued slice-torus invariant q_M cannot be realized as a linear combination of Rasmussen's s-invariant, Ozsváth–Szabó's τ-invariant, all of the 𝔰𝔩_N-concordance invariants (N ≥ 2), Baldwin–Sivek's instanton τ-invariant, Daemi–Imori–Sato–Scaduto–Taniguchi's instanton s̃-invariant and Sano–Sato's Rasmussen type invariants s̃s̃_c.

Khovanov homology and singular instanton Floer homology are both functorial with respect to link cobordisms. Although the two homology groups are related by a spectral sequence, direct correspondence between the cobordism maps has not been rigorously established. In this paper, we define a cobordism map on the instanton cube complex as a filtered chain map, and prove that it recovers the cobordism maps both in Khovanov homology and singular instanton theory. In a sequel paper, we further extend this cobordism map to immersed cobordisms.

We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements $c$ in any principal ideal domain $R$. For the special case $(R, c) = (F[H], H)$ where $F$ is any field, we prove that $\tilde{ss}_c$ coincides with the Rasmussen invariant $s^F$ over $F$. Compared with the unreduced invariants $ss_c$ defined by the first author in a previous paper, we prove that $ss_c = \tilde{ss}_c$ for $(R, c) = (F[H], H)$ and $(\mathbb{Z}, 2)$. However for $(R, c) = (\mathbb{Z}, 3)$, computational results show that $ss_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram [Formula: see text] we construct a CW-spectrum [Formula: see text] whose reduced cellular cochain complex gives the Bar-Natan complex of [Formula: see text]. The stable homotopy type of [Formula: see text] is a link invariant and is described as the wedge sum of the “canonical sphere spectra”. We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the [Formula: see text]-invariant.

Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and algebraically. In this paper, we propose an alternative approach: we stay in the classical setup and fix the functoriality by simply adjusting the signs of the morphisms associated to the Reidemeister moves and the Morse moves.

By using grid homology theory, we give an explicit algorithm for computing Ozsvath-Stipsicz-Szabo's $\Upsilon$-invariant and the $d$-invariant of Dehn surgeries along knots in $S^3$. As its application, we compute the two invariants for all prime knots with up to 11 crossings.

We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].

Lee homology (a variant of Khovanov homology) over ℚ possesses the "canonical generators" as its basis. The generators (Lee's classes) [α(D, o)] are constructed combinatorially from an oriented link diagram D, one for each alternative orientation o on D. Let R be an integral domain. There exists a family of link homology theory { H_c(-; R) }_c ∈ R, where Khovanov's theory corresponds to c = 0 and Lee's theory corresponds to c = 2. For each c ∈ R ∖ 0, Lee's classes [α(D, o)] can be defined as elements in H_c(D; R), but when c is not invertible then they do not form a basis; in fact they are divisible by c-powers. We define the c-divisibility k_c(D) of [α(D, o)] with o the given orientation of D. For any link L and its diagram D, we prove that s̅_c(L) := 2k_c(D) + w(D) - r(D) + 1 is a link invariant, where w is the writhe, and r is the number of Seifert circles. We pose the question whether s̅_c coincides with Rasmussen's s-invariant. There are several evidences that support the affirmative answer. For instance, s̅_c is a link concordance invariant, and the Milnor conjecture can be reproved using s̅_c. Also for the special case (R, c) = (ℚ[h], h), our s̅_c actually coincides with s as knot invariants.