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Lecture notes on link homologies and knotted surfaces

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Pith's one-line read The notes argue that link homology theories, via the maps they assign to link cobordisms, are effective invariants that distinguish knotted surfaces in 4-space.

desk verdict A useful, honest set of lecture notes on link homology cobordism maps and knotted surfaces, with one localized reproducibility gap in the self-contained Bar-Natan example. read the letter →

arxiv 2507.15305 v1 pith:SNDE3OVR submitted 2025-07-21 math.GT

Kyle Hayden This is my paper · ORCID
classification math.GT MSC 57K1857K10
keywords linkhomologyKhovanovFloercobordismmapsknottedsurfacesslicedisksexoticBar-Natan
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

These lecture notes aim to install one working picture: link homology theories are functors from the category of oriented links and link cobordisms to modules, so a surface $\Sigma$ in $S^3 \times I$ with boundary links $L_0$ and $L_1$ induces a map between the homology groups of those links. For Khovanov homology, the induced map is bigraded of degree $(0, \chi(\Sigma))$, well-defined up to sign, and invariant under isotopy rel boundary; for link Floer homology, decorated cobordisms induce well-defined invariant maps. The payoff is 4-dimensional: these maps distinguish slice disks that share a boundary knot, detect exotically knotted surfaces, and answer a long-standing question by showing that equal-genus Seifert surfaces for a knot need not be isotopic in the 4-ball. The notes then make the machinery hands-on, computing Khovanov cobordism maps through the cube of resolutions and the Bar-Natan category, so the reader can reproduce the applications by hand or with software.

What carries the argument

The load-bearing object is the cobordism map. For Khovanov homology it is assembled from the cube of resolutions: each link diagram gives $2^n$ smoothings, each circle is assigned the rank-two Frobenius algebra $A=R\langle 1,x\rangle$, and the elementary cobordisms---birth, merging saddle, splitting saddle, death, and Reidemeister moves---are converted into the maps $\iota$, $m$, $\Delta$, $\epsilon$, and the local cobordisms encoded in the notes' tables. For link Floer homology, the analogous maps come from counting holomorphic Whitney disks in the symmetric product $\mathrm{Sym}^g(\Sigma)$ that avoid, or intersect with multiplicity, the basepoint divisors. The Bar-Natan category, the category of formal complexes of planar tangles modulo the sphere relation, the torus relation, and the 4-tube relation, is what makes the Khovanov complex and its cobordism maps invariant: invariance under Reidemeister moves is proved by chain homotopy equivalences whose chain homotopies use exactly those local relations.

What would settle it

A direct chain-level calculation of the two Khovanov cobordism maps for the $9_{46}$ slice-disk pair depicted in the notes would settle the distinguishing claim: the notes state one map sends the class $\phi$ to $1$ and the other to $0$, so equal images would refute it. Independently, an explicit isotopy rel boundary between two movies of the same link cobordism whose induced Khovanov maps differ by more than a sign would falsify the central invariance theorem.

Watch

Extended reading notes

Core claim

The central claim is functoriality. A link cobordism $\Sigma \subset S^3 \times I$ between links $L_0$ and $L_1$ induces a map $\mathrm{Kh}(\Sigma): \mathrm{Kh}(L_0) \to \mathrm{Kh}(L_1)$ that is bigraded of degree $(0,\chi(\Sigma))$, well-defined up to sign, and invariant under isotopy of $\Sigma$ rel boundary; the corresponding statement for link Floer homology is that every decorated link cobordism induces a well-defined, isotopy-invariant map $\mathrm{HFL}(\Sigma)$. The notes use these maps as the organizing tool for a survey of knotted surfaces: they distinguish pairs of slice disks with the same boundary, detect exotically knotted surfaces in $B^4$, show that Seifert surfaces of equal genus need not be smoothly isotopic in the 4-ball, and bound stabilization distance. The second half of the notes builds the Khovanov side from scratch---cube of resolutions, the Frobenius algebra $A = R\langle 1,x\rangle$, merge/split/birth/death maps, and the Bar-Natan category of formal complexes modulo local relations---so that the cobordism maps can be computed explicitly.

Load-bearing premise

The whole link Floer half of the survey assumes, as a black box, that the holomorphic-curve counts defining the homology are well-defined and independent of the auxiliary choices; the notes cite rather than prove this analytic foundation.

Editorial extensions

If this is right

  • Slice disks with a common boundary knot that are non-isotopic rel boundary are distinguished by their Khovanov and link Floer cobordism maps (Theorems 1.8 and 1.9).
  • Exotically knotted surfaces in $B^4$ are detected: infinitely many knots bound infinitely many genus-one surfaces that are pairwise topologically isotopic but not diffeomorphic, and Khovanov homology detects such pairs in every genus (Theorems 1.10 and 1.11).
  • Equal-genus Seifert surfaces for a fixed knot need not be smoothly isotopic through surfaces in $B^4$, settling a long-standing question in the negative (Theorem 1.13).
  • The stabilization distance between exotically knotted slice disks can be made arbitrarily large, and Bar-Natan homology can distinguish surfaces even after one internal stabilization (Theorems 1.14 and 1.15).
  • Because the maps compose functorially, composing a distinguishing pair of cobordisms with a ribbon concordance preserves the distinction, so the detection propagates to new surfaces (Exercise 3.27 and the remarks around it).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the notes, the explicit chain-level recipes for Reidemeister and saddle maps suggest that distinguishing slice disks by Khovanov maps can be automated: one could compute the images of the two $9_{46}$ disks over $\mathbb{Z}$ with a program and turn the existence proof into a routine check.
  • Beyond the notes, the invariance proof through the Bar-Natan category indicates that any Frobenius algebra satisfying the same local relations yields a functorial link homology theory, so new theories built this way would immediately come with cobordism maps.
  • Beyond the notes, the link Floer stabilization-distance bounds leave open whether Khovanov or Bar-Natan maps can also force arbitrarily large stabilization distance, a question the notes do not answer.
  • Beyond the notes, the mostly zero or formulaic behavior on closed surfaces suggests that the predictive power of these functors is concentrated in surfaces with boundary; designing a closed-surface invariant would require adding structure beyond the maps surveyed here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. These lecture notes, from the 2024 Georgia Topology Summer School, survey the use of link homology theories — principally Khovanov homology and link Floer homology — as functors from the category of links and link cobordisms in S^3×I to modules, and their applications to knotted surfaces in the 4-ball. The first lecture states the functoriality theorems (Theorems 1.2 and 1.4) and surveys applications: distinguishing slice disks, detecting exotically knotted surfaces, resolving Livingston's Seifert-surface question, and lower bounds on stabilization distance. The second lecture develops Khovanov homology from the cube of resolutions and the Frobenius algebra A = R⟨1,x⟩, discusses the cobordism maps with explicit Reidemeister I and II tables, and includes many worked examples, culminating in a detailed calculation distinguishing two slice disks for the knot 9₄₆. The third lecture introduces Bar-Natan homology over F2[H] and Bar-Natan's formal-complex category, sketches the proof of Reidemeister invariance via local relations, and proves (Proposition 3.7) that a pair of slice disks for a strongly invertible knot remain distinct after one internal stabilization, using a spectral-sequence computation attributed to the program KnotJob. The notes close with the ribbon-concordance application.

Significance. As an expository contribution, these notes are valuable: the survey portions are accurate and carefully attributed, the computations in Figures 21, 23, 25 and Tables 1–2 are explicit enough to serve as worked examples, and the exercises (1.16–1.19, 2.12–2.19, 3.3–3.27) genuinely guide the reader through substantial results from the literature. The notes are unusually candid about provenance, flagging where results are quoted from refereed sources, where proofs are sketched, and where computations come from the computer program KnotJob. The load-bearing claims of the survey — functoriality of Khovanov and link Floer cobordism maps and the applications that follow — rest on cited literature and are reported faithfully. The main caveat is that the notes' own self-contained demonstration (Proposition 3.7) depends on a spectral-sequence step that is not justified in the text and on a computation that is not shipped; this is the one point that needs substantive work before the notes can serve as a fully reliable self-study reference.

major comments (2)
  1. [§3.1, Proposition 3.7 (proof of H·δBN ≠ 0)] The decisive claim H·δBN ≠ 0 is not established by the argument as written. The text states that 'all elements in bigrading (0,1) survive to the second page, which implies that H·δBN ≠ 0,' but survival to the E2 page does not by itself imply the nonvanishing of H·δBN: one must also know that the E∞-class corresponding to δBN is a generator of the free F2[H]-tower summand in BN^{0,1}(K) rather than a torsion class, that the spectral sequence collapses at E3 (the collapse is asserted but the d2 differential is not shown in Table 3), and that the F2[H]-module structure of BN(K) is exactly two free towers plus torsion. None of these identifications is supplied, and δBN itself is never identified among the classes counted in Table 3. Because this is the load-bearing step for Proposition 3.7 — the notes' own demonstration of the stabilization-distance phenomenon — the proof should either be completed (defining the Bar-Natan–Lee–Turner spectral sequence, stating how its E∞ term relates to the associated graded of the H-filtration, and proving the free-tower lemma) or the notes should explicitly defer the verification to [Hay23, §5.2] rather than presenting the implication as established.
  2. [§3.1, Table 3 and the KnotJob computation] The computation behind Table 3 is not reproducible from the notes: the KnotJob input (a machine-readable diagram of the knot K of Figure 32) and the program output are not provided, Figure 32 shows K only as a diagram, and Table 3 gives only the bigraded ranks of part of the first two pages. Consequently the assertions that 'this spectral sequence collapses on its third page' and that 'all elements in bigrading (0,1) survive to the second page' cannot be checked by a reader. Since Proposition 3.7 is advertised as illustrating the power of Bar-Natan homology, the notes should either supply the data as ancillary files or state clearly that the example is an illustration whose computational verification is deferred to the published source [Hay23, §5.2].
minor comments (6)
  1. [§2.1, definition of the quantum grading] The two displayed expressions defining q(α) are mutually inconsistent when n− ≠ 0: 'v+(α) − v−(α) + h(α) + n+ − n−' is not equal to 'deg(α) + h(α) + n+ + n−'; the intended quantum-grading convention should be stated correctly.
  2. [Example 1.5] Example 1.5 uses U for the Bar-Natan variable, while the rest of the notes (e.g., §3.1) fixes R = F2[H]; the notes should adopt a single notation for this variable or explicitly note the change of convention.
  3. [§1.5, machine computation] The sentence 'efficient software exists to compute the hat-flavored knot Floer homology [HFK of knots with > 100 crossings' contains an unclosed bracket, presumably a typo for '[HFK]'; the citation formatting should be fixed.
  4. [Table 3, caption] The caption does not state what the entries of Table 3 are; it should say that the entries are bigraded ranks over F2 and should identify which rows and columns are omitted and why those ranges suffice for the argument.
  5. [§3.1, proof of Proposition 3.7] In the proof sketch, the verification that φ is mapped to 1 by Kh(−D) and to 0 by Kh(−D′) is left to the reader as an exercise in the middle of a proof; a brief indication of the movie (for example, which band moves occur) would make the sketch practicable.
  6. [References, [Ozs06]] The citation key [Ozs06] is used in the text and in the reference list for a paper whose authors are Ozsváth and Szabó; the key should be [OS06] (or similar) for consistency with the surrounding OS citations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the notes survey external theorems; Proposition 3.7 has a verification gap (unshipped KnotJob computation) but no definitional or fitted-input circularity.

full rationale

This is a survey/lecture-note paper with no fitted parameters, no data-fitting, and no construction that defines an output in terms of the target. The functoriality theorems (1.2, 2.7, 1.4) and all surveyed applications (1.8–1.15) are quoted from prior published work (Jacobsson, Bar-Natan, Khovanov, Morrison–Walker–Wedrich; Juhasz–Zemke; Sundberg–Swann; Juhasz–Miller–Zemke; Hayden–Sundberg; HKM+22; Guth; Hayden), and the notes do not re-derive those theorems from weaker assumptions. Heavy self-citation (Theorems 1.11, 1.13, 1.15, and examples from [HS24, Hay23, GHKP23]) is present but is not load-bearing in a circular sense: the results are external theorems with independent published proofs, and the notes merely survey them. The only internally presented 'proof' that might be questioned is Proposition 3.7, whose decisive claim H·δBN ≠ 0 is justified by an unshipped KnotJob computation and a terse spectral-sequence inference (Section 3.1, Table 3). That is a verification/completeness gap: the diagram of K, the KnotJob input, and the program output are not provided, and the statement that 'all elements in bigrading (0,1) survive to the second page' does not by itself establish H·δBN ≠ 0 without identifying δBN among the surviving classes and checking later differentials. This is not a circularity: nothing in the argument is defined in terms of the conclusion, and no fitted or computed quantity is renamed as a prediction. Under the stated rules, missing computation and omitted proof support belong to correctness risk rather than circularity, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper is a survey, so its load-bearing input is the existing literature: functoriality theorems for Khovanov and link Floer cobordism maps, the analytic foundations of Heegaard Floer homology, Carter-Saito movie moves, and Bar-Natan's local relations. The only paper-specific input is the KnotJob spectral sequence output in Table 3. There are no free parameters and no invented entities.

assumptions (6)
  • domain assumption Analytic foundations of Heegaard Floer homology: holomorphic Whitney disk moduli spaces in Sym^g(Σ) admit well-defined counts (compactness, transversality) independent of complex structure and Heegaard diagram.
    Invoked in Section 1.1 in the description of the hat and minus flavors of link Floer homology; the notes treat this as a black box (Figure 3 and surrounding discussion).
  • domain assumption Functoriality theorems: Khovanov cobordism maps are invariant up to sign (Theorem 1.2 and Theorem 2.7) and link Floer cobordism maps are invariant for decorated cobordisms (Theorem 1.4).
    Quoted from [Jac04, BN05, Kho06, MWW22] and [JZ20, Zem19b, JMZ21]; the notes only sketch the movie-move strategy in Section 3.2.3.
  • standard math Carter-Saito movie moves: any two movies of isotopic cobordisms are related by finitely many movie moves.
    Invoked in Section 3.2.3 as the framework for proving isotopy invariance of cobordism maps; taken from [CS98].
  • domain assumption Bar-Natan's local relations (S), (T), (4Tu) define the quotient Cob3/l and make the formal complex JTK an invariant (Theorem 3.18).
    Section 3.2.2; the notes verify the Reidemeister I chain homotopy by hand and cite [BN05] for the remaining moves and the planar algebra globalization.
  • domain assumption Conway-Powell theorem: smooth, properly embedded disks in B^4 with common boundary and Z-complement are topologically isotopic rel boundary.
    Used in Theorem 1.12 to certify that the pairs distinguished by [HS24] are exotically knotted rather than merely distinct; quoted from [CP21].
  • ad hoc to paper Correctness of the Bar-Natan-Lee-Turner spectral sequence computation for the knot K of Figure 32, as produced by KnotJob and reported in Table 3.
    Section 3.1, proof of Proposition 3.7: the F2[H]-module structure and the survival of all (0,1)-graded elements to the second page are taken from software output that is not shipped.

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Pith. "Pith review of Lecture notes on link homologies and knotted surfaces." pith.science (2026). https://pith.science/paper/SNDE3OVR

@misc{pith2026250715305,
  author       = {Pith},
  title        = {Pith review of: Lecture notes on link homologies and knotted surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNDE3OVR}},
  note         = {Machine review of arXiv:2507.15305}
}
read the original abstract

Link homology theories (such as knot Floer homology and Khovanov homology) have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions. These notes, based on lectures given at the 2024 Georgia Topology Summer School, discuss what these toolkits say about surfaces in 4-space themselves, via the homomorphisms assigned to link cobordisms. We begin with a brief overview of these theories (focusing on their shared formal properties) and survey some of their applications to knotted surfaces. Afterwards, we give an introduction to Khovanov homology (with an eye towards its cobordism maps), discuss hands-on computational techniques for Khovanov and Bar-Natan homology, and outline the role of the Bar-Natan category in this story.

Figures

Figures reproduced from arXiv: 2507.15305 by the authors.

Figure 1
Figure 1. The cube of resolutions for a diagram of the trefoil. diagram (called a smoothing) with a copy of A⊗k , where k is the number of connected components in the resolved diagram and A is a certain rank-2 Frobenius algebra A = R⟨1, x⟩. Therefore, one may view the generators of CKh(D) as smoothings of D where each connected component is labeled with a choice of 1, x ∈ A. The differential on CKh(D) is defined so that the e… view at source ↗
Figure 2
Figure 2. A doubly-pointed Heegaard diagram for the trefoil (left) and a depiction of the corresponding trefoil (right). (i) (Σ, α, β) is a Heegaard diagram for S 3 with collections of compressing curves α = (α1, . . . , αg) and β = (β1, . . . , βg) for g = g(Σ), (ii) w and z are basepoints in Σ \ (α ∪ β), (iii) K is obtained from a union of (oriented) arcs a ⊂ Σ \ α and b = Σ \ β, where a goes from w to z and b goes from z t… view at source ↗
Figure 3
Figure 3. Schematic depiction of a holomorphic Whitney disk from x to y, including boundary conditions. −i i x Tα Tβ y In the hat flavor, the differential only counts holomorphic Whitney disks that avoid the subspaces {w} × Symg−1 (Σ) and {z} × Symg−1 (Σ). This restriction is dropped in the minus flavor’s chain complex CFL− over R = F2[U, V ], where Whitney disks’ intersections with {w} × Symg−1 (Σ) and {z} × Symg−1 (Σ) are i… view at source ↗
Figures from the paper (34 more)
Figure 4
Figure 4. Figure 4: Decomposing a cobordism into elemen￾tary pieces. (i) birth (ii) saddles (either merging or splitting) (iii) death (iv) Reidemeister I, II, or III moves (v) planar isotopy. An example is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: From left to right: birth, merging saddle, splitting saddle, death, and cylinder [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: A movie of a link cobordism from the empty link to the left-handed trefoil. ≈ [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: The time-reversed mirror of the movie from [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: An example of the swallow-follow isotopy that underlies roll-spinning. Theorem 1.8 (Juh´asz–Zemke [JZ20]). The cobordism maps on link Floer homology can distinguish pairs of slice disks D, D′ ⊂ B4 bounded by the same knot K ⊂ S 3 , up to isotopy rel boundary. The first…
Figure 10
Figure 10. Figure 10: The knot 946, along with a pair of slice disks that it bounds [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: A schematic depiction of rim surgery along a curve γ in an embedded surface [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Rotation of a knotted arc in B3 as applied in twisted rim surgery [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 13
Figure 13. Figure 13: The positron knot and a pair of exotically knotted slice disks that it bounds. Different flavors of exotic surfaces were detected with Khovanov homology: Theorem 1.11 (Hayden–Sundberg [HS24]). For all g ≥ 0, there are infinitely many knots in S 3 that each bound a pai…
Figure 14
Figure 14. Figure 14: (a) The standard Seifert surface for the right-handed trefoil 31. (b-c) Seifert surfaces for the positive Whitehead double Wh(31). show that, over Z2 coefficients, the associated cobordism maps satisfy Kh(Σ) ̸= 0 but Kh(Σ′ ) = 0. See Exercise 2.19 for more. The operat…
Figure 15
Figure 15. Figure 15: A Seifert surface for the trefoil 31 (left), a non-orientable spanning surface for 31 (middle), and a slice disk bounded by the pretzel knot P(−3, 3, −3), also called m(946) (right). 61 820 [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: Band moves encoding slice disks for the knots 820 and 61 [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: Crossing resolutions, and a saddle cobordism between them. 1 2 3 1 1 1 x x x 1 x 1 x 1 x 1 x 1 x 1 x x 1 1 x 1 x 1 x x 1 1 x 1 x 1 x x 1 1 x 1 1 1 x 1 1 1 1 x 1 x 1 x x 1 1 x x x 1 x x x x 000 100 010 m ∆ m ∆ ∆ ∆ ∆ 1 3 2 001 110 101 011 111 ∆ ∆ m ∆ ∆ [PITH_FULL_IMAGE…
Figure 18
Figure 18. Figure 18: The cube of resolutions for a diagram of the trefoil, with a choice of enumer￾ation of the crossings. Smoothings are labeled with their binary coordinates in {0, 1} n=3 and arrows are labeled as merges (m) or splits (∆). Chain complex. Viewing the cube of resolutions …
Figure 19
Figure 19. Figure 19: Examples of generators in CKh(31). Example 2.4. Consider the chain elements α1, . . . , α5 ∈ CKh(31) depicted in Fig￾ure 19; here we use light gray arcs to indicate 0-resolutions. The chains α1, α3, α4, and α5 are cycles, whereas α2 is not. Observe that α3 is the boun…
Figure 20
Figure 20. Figure 20: Saddle, birth, and death cobordisms, along with ornaments , , and . Here we follow Bar-Natan’s convention of reading such diagrams top-to-bottom [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: Calculating the map induced by a standardly embedded torus. Example 2.8. For a standardly embedded torus, viewed as a cobordism T 2 : ∅ → ∅, we can calculate the induced map Kh(T 2 ) : Z → Z as in [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: ■ This brings us to the introduction of one additional ornament: We use a black square ■ on a strand to denote the cobordism obtained by locally summing the product cobordism with a torus. Exercise 2.10. Compute the local effect of this cobordism. In particular, assum…
Figure 23
Figure 23. Figure 23: Calculating the map induced by the standard Seifert surface for −31 (viewed ∅ → −31). is downward. The final column of each table presents the cobordism in a shorthand notation from [BN05]. Here the use of the ornaments , , and ■ is self-explanatory, and the shorthand…
Figure 24
Figure 24. Figure 24: Calculating the image of ϕ ∈ Kh(−946) under the map induced by −D′ [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]
Figure 25
Figure 25. Figure 25: Calculating the image of ϕ ∈ Kh(−946) under the map induced by −D. distinguish the maps induced by −D and −D′ because ψ is symmetric with respect to the symmetry that relates −D and −D′ . Exercise 2.15. Show that the maps induced by −D and −D′ both send ψ to ±1. To fi…
Figure 26
Figure 26. Figure 26: Producing the class ψ ∈ Kh(−946). considering the movie descriptions of the disks and the maps induced by Reidemeis￾ter moves and band moves. First, if we want a class to map nontrivially under the cobordism map, it must satisfy the following: • any crossing that will…
Figure 27
Figure 27. Figure 27: Surfaces Σ (left) and Σ′ (right) bounded by the 3-component unlink [PITH_FULL_IMAGE:figures/full_fig_p026_27.png]
Figure 28
Figure 28. Figure 28: Representing the knot 10148 as the closure of a quasipositive braid, along with the corresponding chain representative of Plamenevskaya’s invariant. Exercise 2.17. (Plamenevskaya’s invariant) Let L be an oriented link expressed as the closure of an n-stranded braid β …
Figure 29
Figure 29. Figure 29: The 2-copy of the right-handed trefoil (top) and the positive Whitehead double of the right-handed trefoil (bottom) [PITH_FULL_IMAGE:figures/full_fig_p028_29.png]
Figure 30
Figure 30. Figure 30: A cycle θ in BN(−946). Due to the additional terms in the differential, it is typically more difficult to explicitly identify cycles at the chain level in Bar-Natan homology. For example, while the labeled smooth￾ings ϕ and ψ from Figures 24 and 26 are cycles when vie…
Figure 31
Figure 31. Figure 31: The unknot, equipped with two different orientations. The next exercise considers the effect of local internal stabilization on cobordism maps in Bar-Natan homology. Exercise 3.5. In contrast with Exercise 2.10, show that BN(Σ#T 2 ) = H · BN(Σ). In fact, the same hold…
Figure 32
Figure 32. Figure 32: A strongly invertible knot (K, τ ) bounding a slice disk D, along with class ϕ ∈ Kh(−K) that distinguishes −D and −τ (D) up to isotopy rel boundary. Distinguishing surfaces via Bar-Natan homology. Considering the case of link cobordisms Σ : K → ∅, the above properties…
Figure 33
Figure 33. Figure 33: A pair of 1-manifolds σ, σ′ and a cobordism Σ between them [PITH_FULL_IMAGE:figures/full_fig_p034_33.png]
Figure 34
Figure 34. Figure 34: Six elementary cobordisms that generate the morphisms in Cob3 . Definition 3.8. Define a functor F : Cob3 → ModR as follows: • Each circle is assigned a free R-module A = R⟨1, x⟩. • Each closed 1-manifold σ is assigned the tensor product Aσ = A⊗|σ| of the copies of A …
Figure 35
Figure 35. Figure 35: Invariance under one of the Reidemeister I moves. Theorem 3.18 ([BN05, Theorem 1]). The isomorphism class of the complex JTK, viewed in Kob/h, is an invariant of the tangle T. The strategy of the proof is to associate chain homotopy equivalences to each of the three R…
Figure 41
Figure 41. Figure 41: The neck-cutting relation in the Bar-Natan category. Bar-Natan then shows that any two local movies that are related by movie moves induce the same morphisms in Kob/h (up to multiplication by ±1). In some cases, this is achieved by directly calculating the induced mor…
Figure 42
Figure 42. Figure 42: Relations in Bar-Natan’s dotted cobordism category. We close with an application to ribbon concordance. Given a pair of knots K and K′ in S 3 , recall that a concordance from K to K′ is a smoothly embedded annulus C ⊂ S 3 × [0, 1] cobounded by K ⊂ S 3 × 0 and K′ ⊂ S 3…
Figure 43
Figure 43. Figure 43: (Left) The knot 811, drawn to suggest a ribbon concordance C from the right-handed trefoil knot to 811. (Right) The double C¯ ◦ C. References [Ali19] Alishahi, Unknotting number and Khovanov homology, Pacific J. Math., 301(1):15–29, 2019. [BN05] Bar-Natan, Khovanov’s …

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