Ribbon distance and Khovanov homology

Sucharit Sarkar

arXiv:1903.11095·math.GT·April 29, 2020

Ribbon distance and Khovanov homology

Sucharit Sarkar

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TL;DR

This paper introduces a new way to measure how close two knots are using ribbon concordances and establishes a lower bound on this distance via Khovanov homology's X-action.

Contribution

It develops a novel lower bound on ribbon distance between knots based on Khovanov homology, linking knot concordance and homological invariants.

Findings

01

Lower bound on ribbon distance using Khovanov homology.

02

Connection between ribbon concordances and homological invariants.

03

Application of Lee's perturbation in bounding knot distances.

Abstract

We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the X-action on Lee's perturbation of Khovanov homology.

Equations44

(2 X)^{d} Kh_{L} (K) ≅ (2 X)^{d} Kh_{L} (K^{'})

(2 X)^{d} Kh_{L} (K) ≅ (2 X)^{d} Kh_{L} (K^{'})

1 \mapsto 1 \otimes X + X \otimes 1, X \mapsto X \otimes X + T 1 \otimes 1.

1 \mapsto 1 \otimes X + X \otimes 1, X \mapsto X \otimes X + T 1 \otimes 1.

Kh_{L} (F) : Kh_{L} (K_{0}) \to Kh_{L} (K_{1})

Kh_{L} (F) : Kh_{L} (K_{0}) \to Kh_{L} (K_{1})

Kh_{L} (K # K^{'}) ≅ Σ^{0, - 1} Tor_{R [X]} (Kh_{L} (K), Kh_{L} (K^{'}))

Kh_{L} (K # K^{'}) ≅ Σ^{0, - 1} Tor_{R [X]} (Kh_{L} (K), Kh_{L} (K^{'}))

K

K

CKh_{L} (K # K^{'}) ≅ Σ^{0, - 1} CKh_{L} (K) \otimes_{R [X]} CKh_{L} (K^{'}) .

CKh_{L} (K # K^{'}) ≅ Σ^{0, - 1} CKh_{L} (K) \otimes_{R [X]} CKh_{L} (K^{'}) .

CKh_{L} (K) \otimes_{R [T]} CKh_{L} (K^{'}) ≅ CKh_{L} (K ⨿ K^{'}) \to Σ^{0, 1} CKh_{L} (K # K^{'}),

CKh_{L} (K) \otimes_{R [T]} CKh_{L} (K^{'}) ≅ CKh_{L} (K ⨿ K^{'}) \to Σ^{0, 1} CKh_{L} (K # K^{'}),

CKh_{L} (K) \otimes_{R [X]} CKh_{L} (K^{'}) \to Σ^{0, 1} CKh_{L} (K # K^{'})

CKh_{L} (K) \otimes_{R [X]} CKh_{L} (K^{'}) \to Σ^{0, 1} CKh_{L} (K # K^{'})

T (K) = {a \in Kh_{L} (K) ∣ \exists n, X^{n} a = 0},

T (K) = {a \in Kh_{L} (K) ∣ \exists n, X^{n} a = 0},

T (K # K^{'}) ≅ Σ^{0, s (K^{'})} T (K) \oplus Σ^{0, s (K)} T (K^{'}) \oplus Σ^{0, - 1} Tor_{F [X]} (T (K), T (K^{'})),

T (K # K^{'}) ≅ Σ^{0, s (K^{'})} T (K) \oplus Σ^{0, s (K)} T (K^{'}) \oplus Σ^{0, - 1} Tor_{F [X]} (T (K), T (K^{'})),

xo (K # K^{'}) = max {xo (K), xo (K^{'})} .

xo (K # K^{'}) = max {xo (K), xo (K^{'})} .

Kh_{L} (L) ≅ Σ^{0, s (L) + 1} F [X] \oplus T (L)

Kh_{L} (L) ≅ Σ^{0, s (L) + 1} F [X] \oplus T (L)

xo (K # K^{'}) \geq max {xo (K), xo (K^{'})}

xo (K # K^{'}) \geq max {xo (K), xo (K^{'})}

K ⟶ F_{2} K ⟶ F_{3} K^{'} ⨿ U^{d} ⟶ \overline{F}_{3} K ⟶ \overline{F}_{2} K .

K ⟶ F_{2} K ⟶ F_{3} K^{'} ⨿ U^{d} ⟶ \overline{F}_{3} K ⟶ \overline{F}_{2} K .

K ⟶ F_{2} K ⟶ F_{3} K^{'} ⨿ U^{d} ⟶ Id K^{'} ⨿ U^{d} ⟶ \overline{F}_{3} K ⟶ \overline{F}_{2} K .

K ⟶ F_{2} K ⟶ F_{3} K^{'} ⨿ U^{d} ⟶ Id K^{'} ⨿ U^{d} ⟶ \overline{F}_{3} K ⟶ \overline{F}_{2} K .

\big{(}K\stackrel{{\scriptstyle F_{2}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle F_{3}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle F_{4}}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}_{4}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle\overline{F}_{3}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle\overline{F}_{2}}}{{\longrightarrow}}K\big{)}=K\stackrel{{\scriptstyle F}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}}}{{\longrightarrow}}K

\big{(}K\stackrel{{\scriptstyle F_{2}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle F_{3}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle F_{4}}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}_{4}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle\overline{F}_{3}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle\overline{F}_{2}}}{{\longrightarrow}}K\big{)}=K\stackrel{{\scriptstyle F}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}}}{{\longrightarrow}}K

Kh_{L} (K) ≅ Σ^{0, s (K) + 1} F [X] \oplus T (K) .

Kh_{L} (K) ≅ Σ^{0, s (K) + 1} F [X] \oplus T (K) .

X^{d} Kh_{L} (K) ≅ Σ^{0, s (K) + 1 - 2 d} F [X] \oplus X^{d} T (K) ≅ X^{d} Kh_{L} (U) ≅ Σ^{0, 1 - 2 d} F [X],

X^{d} Kh_{L} (K) ≅ Σ^{0, s (K) + 1 - 2 d} F [X] \oplus X^{d} T (K) ≅ X^{d} Kh_{L} (U) ≅ Σ^{0, 1 - 2 d} F [X],

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Full text

Ribbon distance and Khovanov homology

Sucharit Sarkar

[email protected]

Department of Mathematics, University of California, Los Angeles, CA 90095

Abstract.

We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the $X$ -action on Lee’s perturbation of Khovanov homology.

SS was supported by NSF Grant DMS-1643401.

1. Introduction

Ever since its inception, Khovanov homology [Kho00], a categorification of the Jones polynomial, has attracted tremendous interest and has produced an entire new field of research. It has been generalized in several orthogonal directions [Bar05, Kho02, Lee05, KR08] and continues to generate intense activity. While the primary focus of the field has been categorification of various low-dimensional topological invariants—endowing them with new algebraic and higher categorical structure—it has also produced a small number of stunning applications in low-dimensional topology as a by-product. Specifically, Lee’s perturbation of Khovanov homology has been instrumental in producing several applications for knot cobordisms; the author’s personal favorites are Rasmussen’s proof of the Milnor conjecture [Ras10] (bypassing the earlier gauge-theoretic proof by Kronheimer-Mrowka) and Piccirillo’s proof that the Conway knot is not slice [Pic].

This is a very short paper, so we will not burden it with a long introduction. Let us quickly describe the main results, and proceed onto the next section.

We define a notion of distance between knots, using the number of saddles in ribbon concordances connecting the knots. This distance is finite if and only if the knots are concordant, but it is hard to find examples of knots arbitrarily large finite distance apart. Using the $X$ -action on Lee’s perturbation of Khovanov homology, we construct a lower bound on this distance, which is the main result of this paper.

Theorem 1.1.

If $d$ is the ribbon distance (defined in Section 3) between knots $K,K^{\prime}$ , then

[TABLE]

where $\mathit{Kh}_{\mathscr{L}}$ is Lee’s perturbation of Khovanov homology.

In particular, the distance of $K$ from the unknot defines a notion of complexity for $K$ , and it has a lower bound coming from Khovanov homology. (Coincidentally, this lower bound agrees with the lower bound on unknotting number from [AD].)

Corollary 1.2.

For any knot $K$ , and over any field $\mathbb{F}$ with $2\neq 0$ , the extortion order $\mathrm{xo}(K)$ (defined in Section 5) is a lower bound for the ribbon distance of $K$ from the unknot.

Acknowledgment

There has been a sudden abundance of short cute papers on applications of Khovanov homology to knot cobordisms, and in particular ribbon concordances [AD, MM, LZ]; the present paper is a result of the author’s desire to join the bandsumwagon. Some of the ideas of this paper are recycled from the above-mentioned papers, and therefore, he is grateful to their authors. He would also like to thank Brendan Owens for pointing out some lower bounds for the band number and Robert Lipshitz for suggesting the wordplay in this paragraph.

2. Knot cobordisms

A cobordism from a link $K_{0}\subset\mathbb{R}^{3}\times\{0\}$ to a link $K_{1}\subset\mathbb{R}^{3}\times\{1\}$ is a properly embedded oriented surface $F\subset\mathbb{R}^{3}\times[0,1]$ (i)(i)(i)We are working with cobordisms in $\mathbb{R}^{3}\times[0,1]$ as opposed to the more standard $S^{3}\times[0,1]$ for a couple of reasons: naturality of Khovanov cobordism maps has only been established in $\mathbb{R}^{3}\times[0,1]$ (even up to sign); and a subtle sign discrepancy for dotted cobordism maps can be resolved in $\mathbb{R}^{3}\times[0,1]$ . with boundary the union of $K_{1}$ and the orientation-reversal of $K_{0}$ . Call the projection $\pi_{t}\colon\mathbb{R}^{3}\times[0,1]\to[0,1]$ the time function, and assume $\pi_{t}|_{F}$ is Morse; its index $0,1,2$ critical points are called births, saddles, and deaths. The cobordism may be viewed as a movie as time runs from [math] to $1$ . For regular values $t$ of $\pi_{t}|_{F}$ , $K_{t}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=F\cap(\mathbb{R}^{3}\times\{t\})$ is a link; $K_{t}$ changes by isotopy with time, with the following local modifications occurring at births, saddles, and deaths:

[TABLE]

We usually work with the projection $\pi_{R}\colon\mathbb{R}^{2}\times\mathbb{R}\times[0,1]\to\mathbb{R}^{2}$ , and represent each $K_{t}\subset\mathbb{R}^{3}\times\{t\}$ by the link diagram $\pi_{R}(K_{t})$ . We then represent the cobordism as a movie of link diagrams; usually $\pi_{R}(K_{t})$ changes by planar isotopy with time, with Reidemeister moves and the above moves happening at certain time instances (which by genericity we will assume to be distinct). Two such movies represent isotopic cobordisms (relative $K_{0}$ and $K_{1}$ ) if and only if they are related by a sequence of movie moves [CRS97].

If $F$ is diffeomorphic to a cylinder, then $F$ is said to be a concordance(ii)(ii)(ii)In old literature, the word ‘cobordism’ was used instead of ‘concordance’. from the knot $K_{0}$ to the knot $K_{1}$ . The concordance is said to be ribbon if there are no births [Gor81]. The famous slice-ribbon conjecture states that every slice knot has a ribbon concordance to the unknot.

We will also be interested in dotted cobordisms, that is, cobordisms $F$ decorated with finite number of dots in the interior. We can also represent them by movies of link diagrams, except now dots are present at certain instances. As before, by genericity, we will assume these instances are separate from the births, saddles, deaths, and the Reidemeister moves; moreover, at each such $t$ , the link $K_{t}$ contains exactly one dot and its projection to the link diagram $\pi_{R}(K_{t})$ is away from the crossings.

Lemma 2.1.

Assume $F,F^{\prime}$ are dotted cobordisms (in generic position) with the same underlying surface and the same number of dots on each component, but differing only in the placement of the dots. Then the movies for $F$ and $F^{\prime}$ are related a sequence of the following movie moves.

(1)

Far commutation: We may switch the order of the following operations,

(a)

adding a dot, and then adding another dot; 2. (b)

adding a dot, and then performing a birth; 3. (c)

adding a dot, and then performing a saddle; 4. (d)

performing a death, and then adding a dot; 5. (e)

adding a dot, and then performing a Reidemeister move far away. 2. (2)

Moving dots on link diagrams: If we are adding a dot on one side of a crossing on a link diagram $\pi_{R}(K_{t})$ , then we can instead add it on the other side of the crossing.

Proof.

We can use the second movie move to move dots freely on $K_{t}$ for each $t$ . To move dots in the time direction, we use the first movie move, which allows us to move dots past each other, and also past births, deaths, saddles, and Reidemeister moves; all the possibilities are listed, except the following.

(a)

Perform a birth, and then add a dot on the newborn unknot component. In this case, it is impossible to switch the order. 2. (b)

Add a dot to a small unknot component, and then perform a death on that component. This is simply the time-reversal of the previous case. 3. (c)

Add a dot on some strand of the knot diagram, and then perform a Reidemeister move that involves that strand; see below for an example with Reidemeister II move.

[TABLE]

However, in this case, we may move the dot on the link diagram (using the second movie move) away from the strands involved in the Reidemeister move, and then use far commutation with the Reidemeister move (using the first movie move) to change the temporal order of the dot addition and the Reidemeister move.∎

3. Ribbon complexities

There are certain notions of complexities that we can associate to ribbon concordances. If $K$ is a ribbon knot—that is, if $K$ has a ribbon concordance to the unknot $U$ —then we can define the band number $b(K)$ to be the smallest number of saddles in a ribbon concordance $K\to U$ ; this is also the smallest number of bands if we write $K$ as a band sum of an unlink. This number is usually called the ribbon-fusion number and has lower bounds coming from the Jones polynomial [Kan10]; more classically, it is bounded below by half of $\mathrm{rk}(H_{1}(\Sigma_{K}))$ —the smallest number of generators for the first homology of the double branched cover [NN82].

Each ribbon knot bounds a ribbon disk in $\mathbb{R}^{3}$ , that is, an immersed disk with only ribbon singularities (as shown with the thick line in the leftmost figure below). So we may define the ribbon number $r(K)$ to be smallest number of ribbon singularities for ribbon disks bounding $K$ . We may perform saddles near each ribbon singularity (as shown below) to convert $K$ to an unlink, so $r(K)$ is bounded below by $b(K)$ . A nice argument shows that the knot genus $g(K)$ also provides a lower bound for $r(K)$ [Fox73].

[TABLE]

We may also define a notion of distance on knots coming from ribbon concordances. For any two knots $K,K^{\prime}$ , define the ribbon distance $d(K,K^{\prime})$ to be the smallest $k$ such that there is a sequence of knots $K=K_{0},K_{1},\dots,K_{n-1},K_{n}=K^{\prime}$ from $K$ to $K^{\prime}$ and a ribbon concordance (in some direction) between every consecutive pair $K_{i},K_{i+1}$ with at most $k$ saddles. The following properties are immediate.

(1)

$d(K,K^{\prime})<\infty$ if and only if $K$ and $K^{\prime}$ are concordant. (For the slightly non-obvious direction, note that if $K$ and $K^{\prime}$ are concordant, then there is some $K^{\prime\prime}$ with ribbon concordances to both $K$ and $K^{\prime}$ , cf. [Gor81].) 2. (2)

$d(K,K^{\prime})=0$ if and only if $K$ and $K^{\prime}$ are isotopic. 3. (3)

$d(K,K^{\prime})=d(K^{\prime},K)$ . 4. (4)

$d(K,K^{\prime\prime})\leq\max\{d(K,K^{\prime}),d(K^{\prime},K^{\prime\prime})\}$ , and hence $d$ satisfies the triangle inequality.

This notion of distance complements the more standard notion of cobordism distance which is defined to be the smallest genus of a cobordism between the two knots. (Cobordism distance between any two knots is finite, and is zero if and only if the knots are concordant.)

For any slice knot $K$ , its distance from the unknot, $d(K,U)$ , therefore provides yet another notion of complexity. It is clear from the definitions that $d(K,U)\leq b(K)$ .

*Example 3.1**.*

Let $K_{1}$ be the connect sum of the positive and the negative trefoil, and let $K_{n}$ be the connect sum of $n$ copies of $K_{1}$ . We have $r(K_{1})=g(K_{1})=\mathrm{rk}(H_{1}(\Sigma_{K_{1}}))=2$ and $b(K_{1})=d(K_{1},U)=1$ ; $K_{1}$ can be obtained by adding a band (shown by the thick line below) to the $2$ -component unlink, which intersects the natural disks bounding the unlink in $2$ ribbon singularities.

[TABLE]

We get $r(K_{n})\geq g(K_{n})=ng(K_{1})=2n$ . The ribbon number is sub-additive under connect sum, so $r(K_{n})\leq nr(K_{1})=2n$ ; therefore, $r(K_{n})=2n$ . We also get $b(K_{n})\geq\mathrm{rk}(H_{1}(\Sigma_{K_{n}}))/2=\mathrm{rk}(\oplus^{n}H_{1}(\Sigma_{K_{1}}))/2=n$ . The band number is also sub-additive under connect sum, so $b(K_{n})\leq nb(K_{1})=n$ ; therefore, $b(K_{n})=n$ . Finally $d(K_{n},U)=1$ since we have a sequence of knots $K_{n},K_{n-1},\dots,K_{0}=U$ , and a single-saddle ribbon concordance $K_{i+1}\to K_{i}$ for all $i$ , obtained by connect summing $K_{i}$ with the single-saddle ribbon concordance $K_{1}\to U$ .

It is unclear if $d(K,U)$ can be arbitrarily large (while staying finite). In this paper, we will give an example of a knot with $d(K,U)=2$ (Example 6.1), and indeed one with $d(K,U)>2$ (Example 6.2). It is reasonable to guess that the techniques of this paper, but using knot Floer homology instead of Khovanov homology, might produce examples of knots with larger values of $d(K,U)$ .

4. Khovanov homology

Fix a ground ring $R$ and consider the $2$ -dimensional Frobenius algebra $V=R[T][X]/\{X^{2}=T\}$ over $R[T]$ with comultiplication $V\to V\otimes_{R[T]}V$ given by

[TABLE]

and counit $V\to R[T]$ given by $1\mapsto 0,X\mapsto 1$ . This produces a Khovanov-style link homology theory [Kho00] for any link $K$ by applying it to the Kauffman cube of resolutions of its link diagram. The resulting theory is usually called the Lee perturbation of Khovanov homology [Lee05], and we will denote it $\mathit{Kh}_{\mathscr{L}}(K)$ . It is a bigraded homology theory over $R[T]$ with $R$ in bigrading $(0,0)$ and $T$ in bigrading $(0,-4)$ .

A dotted cobordism $F\colon K_{0}\to K_{1}$ (in generic position) with $\delta(F)$ dots induces a map

[TABLE]

of $R[T]$ -modules of bigrading $(0,\chi(F)-2\delta(F))$ [Kho06, Bar05, Jac04, LZ], defined as follows. The movie presentation for $F$ is a sequence of planar isotopy, Reidemeister moves, births, saddles, deaths, and dot additions. Except dot addition, each of the other moves induce a map on $\mathit{Kh}_{\mathscr{L}}$ using the Frobenius algebra $V$ . The dot addition map is defined slightly differently. We present a careful definition below that avoids a sign issue.

An elementary dotted cobordism from $K\to K$ is a product cobordism decorated with a single dot. Consider (the projection of) the dot on the oriented link diagram $\pi_{R}(K)$ . Checkerboard color the complement of the link diagram in $\mathbb{R}^{2}$ so that the unbounded region is colored white. If the arc in the link diagram containing the dot is oriented as the boundary of a black region, define the sign of the dot to be $(+1)$ , otherwise, define it to be $(-1)$ . Then define the dotted cobordism map $\mathit{Kh}_{\mathscr{L}}(K)\to\mathit{Kh}_{\mathscr{L}}(K)$ to be the map merging a small unknot labeled $X$ near the dot, times the sign of the dot.

It is well-known that two isotopic (rel boundary) undotted knot cobordisms induce the same map $\mathit{Kh}_{\mathscr{L}}(K_{0})\to\mathit{Kh}_{\mathscr{L}}(K_{1})$ , up to an overall sign.(iii)(iii)(iii)This sign issue can also be resolved [CMW09], but we will not need to. We have a similar variant for dotted cobordisms.

Lemma 4.1.

Assume $F,F^{\prime}$ are dotted cobordisms (in generic position) with the same underlying surface and the same number of dots on each component, but differing only in the placement of the dots. Then they induce the same map on $\mathit{Kh}_{\mathscr{L}}$ , including the sign.

Proof.

We merely have to check that the map is unchanged under the movie moves listed in Lemma 2.1. The first movie move (far commutation) is clear. For the second movie move (moving the dot past a crossing), we may check directly that on the Khovanov chain complex level, the map associated to merging a small unknot labeled $X$ to some strand is homotopic to negative of the map associated to merging a small unknot labeled $X$ to the corresponding strand on the opposite side of a crossing, cf. [BLS17]. Therefore we have the same map on homology for dot addition on either side of a crossing. ∎

The main advantage of using dotted cobordisms is the famous neck-cutting relation. We will need it in the following two forms.

Lemma 4.2.

Assume the link diagram $\pi_{R}(K)$ for $K$ contains a small unknot $U$ . Then, up to an overall sign, the identity map $\mathit{Kh}_{\mathscr{L}}(K)\to\mathit{Kh}_{\mathscr{L}}(K)$ is the sum of the following two maps,

(1)

add a dot to $U$ , perform a death on $U$ , and perform a rebirth for $U$ ; 2. (2)

perform a death on $U$ , perform a rebirth for $U$ , and add a dot to $U$ .

In terms of movies,

[TABLE]

Proof.

If the dot addition maps are given by merging small unknots labeled $X$ , then it is easy to check that the above equation holds (without the sign) on the nose at the Khovanov chain complex level. However, the actual dot addition map has an extra sign given by the sign of the dot. But the unknot $U$ before death and the unknot $U$ after birth are oriented in the same way, so the two dots have the same sign, and consequently, the above equation holds up to an overall sign. ∎

Lemma 4.3.

Assume $F\colon K\to K$ is a cobordism obtained by performing an elementary saddle on the link diagram $\pi_{R}(K)$ for $K$ , followed by performing the saddle in reverse. Then, up to an overall sign, the map $\mathit{Kh}_{\mathscr{L}}(K)\to\mathit{Kh}_{\mathscr{L}}(K)$ is the sum of the following two maps,

(1)

add a dot to one of the two strands in $\pi_{R}(K)$ involved in the saddle; 2. (2)

add a dot to the other strand in $\pi_{R}(K)$ involved in the saddle.

In terms of movies,

[TABLE]

Proof.

The proof is very similar to the previous proof. If the dot addition maps are given by merging small unknots labeled $X$ , then the equation holds (without the sign) at the Khovanov chain complex level. However, since the saddle is an oriented saddle, the two dots on the two strands have the same sign, and consequently, the above equation holds up to an overall sign. ∎

5. $X$ -action on Khovanov homology

If we fix a component of $K$ , then the map $\mathit{Kh}_{\mathscr{L}}(K)\to\mathit{Kh}_{\mathscr{L}}(K)$ associated to the elementary dotted cobordism $K\to K$ that has a single dot on the chosen component is denoted $X$ (since it comes from merging an unknot labeled $X$ ), and is often called the $X$ -action on $\mathit{Kh}_{\mathscr{L}}(K)$ . It is clear from the Frobenius algebra $V$ that $X^{2}=T$ . This makes $\mathit{Kh}_{\mathscr{L}}(K)$ a module over $R[T,X]/\{X^{2}=T\}=R[X]$ (although the module structure depends on chosen link component).

Khovanov homology of connect sums has a nice expression using the $X$ -actions.

Lemma 5.1.

Let $K,K^{\prime}$ be links with chosen components, and let $K\#K^{\prime}$ be the link obtained by connect summing the chosen components. Then

[TABLE]

as bigraded $R[X]$ -modules, with the $X$ -action on the right-hand side induced from the $X$ -action on either $\mathit{Kh}_{\mathscr{L}}(K)$ or $\mathit{Kh}_{\mathscr{L}}(K^{\prime})$ . Here $\Sigma^{a,b}$ denotes an upward bigrading shift by $(a,b)$ , that is, tensoring with a single $R$ in bigrading $(a,b)$ .

Proof.

The argument entirely follows Khovanov’s argument for his original invariant (which is the specialization $X^{2}=T=0$ ), so we skip some details. Consider the following link diagrams for $K$ , $K^{\prime}$ , and $K\#K^{\prime}$ , so that the induced diagram for $K\amalg K^{\prime}$ differs from the diagram of $K\#K^{\prime}$ locally by an elementary saddle.

[TABLE]

Let $\mathit{CKh}_{\mathscr{L}}$ be the Khovanov chain complexes associated to these diagrams. They become modules over $R[X]$ by the $X$ -action at the strands that are shown in the above diagram. (For $K\#K^{\prime}$ either strand works.) By construction, these complexes are free over $R[T]$ , but indeed, they are free over $R[X]$ as well. Therefore, it is enough to construct an isomorphism of chain complexes over $R[X]$ ,

[TABLE]

Consider the saddle map

[TABLE]

and it is easy to check that it factors through $\mathit{CKh}_{\mathscr{L}}(K)\otimes_{R[X]}\mathit{CKh}_{\mathscr{L}}(K^{\prime})$ . So all that remains is to check that this $R[X]$ -module chain map

[TABLE]

is an isomorphism on the chain groups.

The chain groups $\mathit{CKh}_{\mathscr{L}}$ decompose as direct sums of chain groups of various resolutions of the link diagrams, so it is enough to check that the above map is an isomorphism at each resolution of $K$ and $K^{\prime}$ —that is, it is enough to check the case when $K$ and $K^{\prime}$ are planar unlinks, which is trivial to check. ∎

If $K$ is a knot, and $R$ is a field $\mathbb{F}$ with $2\neq 0$ , then the module $\mathit{Kh}_{\mathscr{L}}(K)$ over $\mathbb{F}[X]$ takes a particularly simple form. It decomposes (non-canonically) as $\Sigma^{0,s(K)+1}\mathbb{F}[X]\oplus\mathscr{T}(K)$ , where $s(K)$ is Rasmussen’s $s$ -invariant, and $\mathscr{T}(K)$ is the (canonical) subgroup of $\mathit{Kh}_{\mathscr{L}}(K)$ consisting of the $X$ -torsion elements,

[TABLE]

which we will call the extortion group of $K$ .

The smallest $n$ such that $X^{n}\mathscr{T}(K)=0$ is called the extortion order, and denoted $\mathrm{xo}(K)$ . This was used earlier in [AD] to provide a lower bound on the unknotting number. The extortion order $\mathrm{xo}(K)$ is related to the Lee spectral sequence (coming from the filtered chain complex for $\mathit{Kh}_{\mathscr{L}}(K)$ with filtration given by powers of $T$ ) as follows. If the Lee spectral sequence collapses at the $E_{k}$ page, then $\mathrm{xo}(K)\in\{2k-3,2k-2\}$ .

The only knot with $\mathrm{xo}(K)=0$ (that is, $\mathscr{T}(K)=0$ ) is the unknot [KM11]. All other Kh-thin knots have $\mathrm{xo}(K)=1$ ; $8_{19}$ is the first knot with $\mathrm{xo}(K)=2$ . Since the Lee spectral sequence collapses at the $E_{2}$ page for small knots, it is hard to find examples of knots with $\mathrm{xo}(K)>2$ ; the first example of a knot with $\mathrm{xo}(K)>2$ was constructed in [MM].

The extortion order can be computed from the Mathematica package KnotTheory [BM] using the function UniversalKh, the standard reference for which seems to be ‘Scott’s slides’ [Mor]. UniversalKh works over $\mathbb{Q}$ and returns a free resolution of $\mathit{Kh}_{\mathscr{L}}(K)$ over $\mathbb{Q}[X]$ . Each term $t^{a}q^{b}\mathrm{KhE}$ contributes a tower $\mathbb{Q}[X]\langle p\rangle$ with the generator $p$ in bigrading $(a,b)$ , and each term $t^{a}q^{b}\mathrm{KhC[n]}$ contributes a two-step complex $\mathbb{Q}[X]\langle p,q\rangle$ with generators $p,q$ in bigradings $(a-1,b-2n),(a,b)$ and differential $p\mapsto X^{n}q$ . Therefore, the extortion group $\mathscr{T}(K)$ over $\mathbb{Q}$ is isomorphic to the homology of complexes coming from the $\mathrm{KhC}[n]$ terms and the extortion order $\mathrm{xo}(K)$ over $\mathbb{Q}$ is the largest $n$ so that $\mathrm{KhC}[n]$ appears.

The extortion groups and extortion orders behave nicely under connect sums.

Lemma 5.2.

Consider knots $K,K^{\prime}$ and their connect sum $K\#K^{\prime}$ . Then over any field $\mathbb{F}$ with $2\neq 0$ ,

[TABLE]

and

[TABLE]

Proof.

The first statement is immediate from Lemma 5.1 and the isomorphism

[TABLE]

for all knots $L$ .

For the second statement, we immediately get

[TABLE]

from the first two summands in the decomposition of $\mathscr{T}(K\#K^{\prime})$ . So it is enough to prove that the extortion order of the summand $\operatorname{Tor}_{\mathbb{F}[X]}(\mathscr{T}(K),\mathscr{T}(K^{\prime}))$ equals the minimum of the extortion order of $\mathscr{T}(K)$ and $\mathscr{T}(K^{\prime})$ .

Consider free resolutions $\widetilde{\mathscr{T}}(K)$ and $\widetilde{\mathscr{T}}(K^{\prime})$ of the extortion groups over $\mathbb{F}[X]$ . By the classification of finitely generated modules over PID’s, they decompose into a direct sum of 2-step complexes $\mathbb{F}[X]\langle p,q\rangle$ , with the differential given by $p\mapsto\alpha(X)q$ , where $\alpha(X)$ is some power of some irreducible homogeneous polynomial in $X$ . Since $X$ has non-zero bigrading, the only possibilities are $\alpha(X)=X^{n}$ . Each such summand contributes $\mathbb{F}[X]\langle q\rangle/\{X^{n}q=0\}$ in homology, so the extortion orders are the maximum $n$ ’s that appear in such a decomposition.

If $n\geq m$ , then by a simple change of basis, the tensor product of the 2-step complexes $\mathbb{F}[X]\stackrel{{\scriptstyle X^{n}}}{{\longrightarrow}}\mathbb{F}[X]$ and $\mathbb{F}[X]\stackrel{{\scriptstyle X^{m}}}{{\longrightarrow}}\mathbb{F}[X]$ decomposes as $\big{(}\mathbb{F}[X]\stackrel{{\scriptstyle X^{m}}}{{\longrightarrow}}\mathbb{F}[X]\big{)}\oplus\big{(}\mathbb{F}[X]\stackrel{{\scriptstyle X^{m}}}{{\longrightarrow}}\mathbb{F}[X]\big{)}$ , and hence, the extortion order of the summand $\operatorname{Tor}_{\mathbb{F}[X]}(\mathscr{T}(K),\mathscr{T}(K^{\prime}))$ equals $\min\{\mathscr{T}(K),\mathscr{T}(K^{\prime})\}$ . ∎

6. Main theorem

This section is devoted to the proof of the main theorems from Section 1.

Proof of Theorem 1.1.

Since the ribbon distance is defined using a sequence of ribbon concordances, it is enough to do the case when there is a ribbon concordance $K\stackrel{{\scriptstyle F}}{{\longrightarrow}}K^{\prime}$ with at most $d$ saddles. After isotopy, we assume the movie of the cobordism $F$ has the following form.

(1)

First we perform some Reidemeister moves and planar isotopy on $K$ . Since we are free to choose the link diagram for $K$ , we actually do not need this move. 2. (2)

Then we perform $d$ elementary (planar) saddles, one at a time. 3. (3)

Then we perform further Reidemeister moves and planar isotopy. 4. (4)

Then we perform $d$ elementary (planar) deaths, again one at a time. 5. (5)

Then we again perform some Reidemeister moves and planar isotopy to end at $K^{\prime}$ . Once again, since we are free to choose the link diagram for $K^{\prime}$ , we do not need this move.

So the ribbon concordance $K\stackrel{{\scriptstyle F}}{{\longrightarrow}}K^{\prime}$ decomposes as $K\stackrel{{\scriptstyle F_{2}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle F_{3}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle F_{4}}}{{\longrightarrow}}K^{\prime}$ , where the piece $F_{i}$ comes from Item-( $i$ ) above, and $U^{d}$ denotes $d$ -component planar unlink.

Let $K^{\prime}\stackrel{{\scriptstyle\overline{F}}}{{\longrightarrow}}K$ be the cobordism viewed in reverse (which decomposes as $K^{\prime}\stackrel{{\scriptstyle\overline{F}_{4}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle\overline{F}_{3}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle\overline{F}_{2}}}{{\longrightarrow}}K$ ). Let $K\stackrel{{\scriptstyle W}}{{\longrightarrow}}K$ be the cobordism

[TABLE]

We will prove Theorem 1.1 by computing the image of the map $\mathit{Kh}_{\mathscr{L}}(W)$ in two different ways, corresponding to the two sides of the equation in the statement of the theorem.

•

Method 1. The cobordism $\widetilde{K}\stackrel{{\scriptstyle F_{3}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle\overline{F}_{3}}}{{\longrightarrow}}\widetilde{K}$ is isotopic (rel boundary) to the identity cobordism $\widetilde{K}\to\widetilde{K}$ since the cobordism $F_{3}$ corresponds to a link isotopy in $\mathbb{R}^{3}$ , and $\overline{F}_{3}$ is the same isotopy performed in reverse. Therefore, the image of $\mathit{Kh}_{\mathscr{L}}(W)$ is same as the image of the map associated to the cobordism $K\stackrel{{\scriptstyle F_{2}}}{{\longrightarrow}}\widetilde{K}\stackrel{{\scriptstyle\overline{F}_{2}}}{{\longrightarrow}}K$ .

This cobordism performs $d$ planar saddles, and then performs them in reverse. So repeated applications of Lemma 4.3 tells us that the map $\mathit{Kh}_{\mathscr{L}}(\overline{F}_{2})\circ\mathit{Kh}_{\mathscr{L}}(F_{2})$ associated to this cobordism, up to an overall sign, is $2^{d}$ times the map associated to the dotted cobordism $K\stackrel{{\scriptstyle P}}{{\longrightarrow}}K$ , where $P$ is the product cobordism decorated with $d$ dots. (Note, since $P$ is connected, by Lemma 4.1, the map $\mathit{Kh}_{\mathscr{L}}(P)$ is independent of the placement of the $d$ dots on $P$ .) By definition, the image of $\mathit{Kh}_{\mathscr{L}}(P)$ is $X^{d}\mathit{Kh}_{\mathscr{L}}(K)$ ; therefore, the image of the original cobordism map is $(2X)^{d}\mathit{Kh}_{\mathscr{L}}(K)$ .

Schematically (with $d=1$ ):

[TABLE]

•

Method 2. Insert the identity cobordism $K^{\prime}\amalg U^{d}\to K^{\prime}\amalg U^{d}$ into $W$ to get an isotopic cobordism

[TABLE]

The cobordism $K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle\operatorname{Id}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}$ has $d$ necks coming from $U^{d}$ , so repeated applications of Lemma 4.2 tells us that the map $\operatorname{Id}\colon\mathit{Kh}_{\mathscr{L}}(K^{\prime}\amalg U^{d})\to\mathit{Kh}_{\mathscr{L}}(K^{\prime}\amalg U^{d})$ , up to an overall sign, is the sum of $2^{d}$ maps associated to the following $2^{d}$ dotted cobordisms with $d$ dots: Each has the same underlying surface $K^{\prime}\amalg U^{d}\stackrel{{\scriptstyle F_{4}}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}_{4}}}{{\longrightarrow}}K^{\prime}\amalg U^{d}$ which has $d$ death-birth pairs; and the $2^{d}$ dotted cobordisms are obtained by distributing $d$ dots in $2^{d}$ different ways so that each death-birth pair has exactly one dot.

The underlying composed cobordism

[TABLE]

is connected, so by Lemma 4.1, the $2^{d}$ dotted cobordism maps all induce the same map, which is the map $\mathit{Kh}_{\mathscr{L}}(\overline{F})\circ\mathit{Kh}_{\mathscr{L}}(Q)\circ\mathit{Kh}_{\mathscr{L}}(F)$ corresponding to the dotted cobordism $K\stackrel{{\scriptstyle F}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle Q}}{{\longrightarrow}}K^{\prime}\stackrel{{\scriptstyle\overline{F}}}{{\longrightarrow}}K$ , where $K^{\prime}\stackrel{{\scriptstyle Q}}{{\longrightarrow}}K^{\prime}$ is the product cobordism decorated with $d$ dots.

Levine and Zemke has shown [LZ] that the map $\mathit{Kh}_{\mathscr{L}}(F)\circ\mathit{Kh}_{\mathscr{L}}(\overline{F})\colon\mathit{Kh}_{\mathscr{L}}(K^{\prime})\to\mathit{Kh}_{\mathscr{L}}(K^{\prime})$ is $\pm\operatorname{Id}$ ,(iv)(iv)(iv)Actually, they proved it for Khovanov’s specialization $X^{2}=T=0$ , but the proof works in this more general case. and therefore, the map $\mathit{Kh}_{\mathscr{L}}(F)$ is surjective and the map $\mathit{Kh}_{\mathscr{L}}(\overline{F})$ is injective. Consequently, the image of the map $\mathit{Kh}_{\mathscr{L}}(\overline{F})\circ\mathit{Kh}_{\mathscr{L}}(Q)\circ\mathit{Kh}_{\mathscr{L}}(F)$ is isomorphic to the image of $\mathit{Kh}_{\mathscr{L}}(Q)$ , which is $X^{d}\mathit{Kh}_{\mathscr{L}}(K^{\prime})$ . Therefore, the image of the original cobordism map is isomorphic to $(2X)^{d}\mathit{Kh}_{\mathscr{L}}(K^{\prime})$ .

Schematically:

[TABLE]

Proof of Corollary 1.2.

Let $d$ be the ribbon distance of the knot $K$ to the unknot $U$ . We know $\mathit{Kh}_{\mathscr{L}}(U)\cong\Sigma^{0,1}\mathbb{F}[X]$ and since we are working over a field with $2\neq 0$ ,

[TABLE]

(If $d<\infty$ , then $K$ is slice, and hence $s(K)=0$ [Ras10], but we will not need this fact; indeed, this fact will follow from the proof.)

By Theorem 1.1, and since $2\neq 0$ ,

[TABLE]

and therefore, $X^{d}\mathscr{T}(K)=0$ , and hence, $d\geq\mathrm{xo}(K)$ . ∎

*Example 6.1**.*

Let $K$ be the connect sum of $8_{19}$ and its mirror. Using the function UniversalKh from the Mathematica package KnotTheory, we get $\mathrm{xo}(8_{19})=2$ . By Lemma 5.2, $\mathrm{xo}(K)\geq 2$ , and hence by Corollary 1.2, the distance of $K$ from the unknot is at least $2$ . Indeed, adding untwisted (blackboard-framed) bands along the thick lines in the following knot diagram for $K$ converts it to a $3$ -component unlink, so $d(K,U)=2$ .

[TABLE]

*Example 6.2**.*

Let $K_{M}$ be the knot from [MM], and let $K$ be the connect sum of $K_{M}$ and its mirror. Since the Lee spectral sequence for $K_{M}$ collapses at the $E_{3}$ page, we know $\mathrm{xo}(K_{M})\geq 3$ . Once again, by Lemma 5.2, $\mathrm{xo}(K)\geq 3$ , and hence by Corollary 1.2, the distance of $K$ from the unknot is at least $3$ .

Bibliography22

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[AD] Akram Alishahi and Nathan Dowlin, The Lee spectral sequence, unknotting number, and the knight move conjecture , ar Xiv:1710.07875.
2[Bar 05] Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms , Geom. Topol. 9 (2005), 1443–1499. MR 2174270 (2006 g:57017)
3[BLS 17] John A. Baldwin, Adam Simon Levine, and Sucharit Sarkar, Khovanov homology and knot Floer homology for pointed links , J. Knot Theory Ramifications 26 (2017), no. 2, 1740004, 49. MR 3604486
4[BM] Dror Bar-Natan, Scott Morrison, and et al., The Knot Atlas , http://katlas.org/ .
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TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Ribbon distance and Khovanov homology

Abstract.

1. Introduction

Theorem 1.1**.**

Corollary 1.2**.**

Acknowledgment

2. Knot cobordisms

Lemma 2.1**.**

Proof.

3. Ribbon complexities

Example 3.1*.*

4. Khovanov homology

Lemma 4.1**.**

Proof.

Lemma 4.2**.**

Proof.

Lemma 4.3**.**

Proof.

5. XXX-action on Khovanov homology

Lemma 5.1**.**

Proof.

Lemma 5.2**.**

Proof.

6. Main theorem

Proof of Theorem 1.1.

Proof of Corollary 1.2.

Example 6.1*.*

Example 6.2*.*

Theorem 1.1.

Corollary 1.2.

Lemma 2.1.

*Example 3.1**.*

Lemma 4.1.

Lemma 4.2.

Lemma 4.3.

5. $X$ -action on Khovanov homology

Lemma 5.1.

Lemma 5.2.

*Example 6.1**.*

*Example 6.2**.*