Span of the Jones polynomials of certain v-adequate virtual links
aa r X i v : . [ m a t h . G T ] J a n SPAN OF THE JONES POLYNOMIALS OF CERTAIN V-ADEQUATEVIRTUAL LINKS
MINORI OKAMURA AND KEIICHI SAKAIA bstract . It is known that the Kau ff man-Murasugi-Thislethwaite type inequality becomesan equality for any (possibly virtual) adequate link diagram. We refine this condition. As anapplication we obtain a criterion for virtual link diagram with exactly one virtual crossingto represent a properly virtual link.
1. I ntroduction
The Kau ff man-Murasugi-Thislethwaite inequality [4, 9, 10], KMT inequality for short,is known as an e ff ective tool to estimate, and in some cases determine, the minimal crossingnumber of (classical) links in terms of the span of the Jones polynomial (or equivalentlyof the Kau ff man bracket polynomial). The KMT inequality is a strict inequality for somelinks, and the defect is closely related to the Euler characteristics of the Turaev surface [11] (also known as the atom [8]). Thus the KMT inequality has a refined form involvingthe Euler characteristics of the Turaev surfaces (Theorem 4.1; see also [1, 2]). Moreover itis known that the refined KMT inequality becomes an equality for adequate link diagrams.The proofs of these results seem easier than that of the original KMT inequality, and in factthe refined inequality provides a simple proof of the Tait conjecture [11].In this paper we refine this su ffi cient condition for (possibly virtual) link diagram underwhich condition the refined KMT inequality becomes an equality. We moreover introducethe notion of v-adequate link diagrams to be diagrams obtained by virtualizing exactlyone real crossing of some adequate diagrams, and as an application we prove that therefined KMT inequality becomes an equality for v-adequate link diagrams with certaincondition. This means that we can determine the span of the Jones polynomials of anadequate diagram and a v-adequate one obtained from the former, and this allows us toshow that one of the span of the Jones polynomials of these links cannot be divided by four.We therefore obtain a recipe for producing properly virtual links. The “certain condition”is valid for v-adequate diagrams derived from classical adequate diagrams, and hence weindeed have infinitely many examples to which our criterion is applied. These examplescan be seen as generalizations of Kishino’s result [6].This paper is organized as follows. In §2 we review the Kau ff man bracket and theTuraev surface of (possibly virtual) link diagrams. We introduce the notion of (pseudo-)adequacy in §3, and we prove our main theorems in §4.2. P reliminaries We follow the conventions in [5] for virtual link diagrams. Two virtual link diagrams D and D ′ are said to be equivalent if D can be transformed into D ′ by a finite sequence ofReidemeister moves and virtual Reidemeister moves [5, Figure 2].2.1. The Kau ff man bracket polynomial. The transformations of a virtual link diagramin a neighborhood of a real crossing as in Figure 2.1 are called respectively
A-splice and
B-splice . We draw a dotted line at each spliced real crossing as in Figure 2.1, which wecall a connecting arc . Date : January 18, 2021.
B-spliceA-spliceF igure F ′ ( D ) virtual crossing F ′ ( D )F igure F ′ ( D )Let D be a virtual link diagram. A state s of D is a map from the set of the real crossingsof D to the set { A , B } . We denote by S the set of the states of D . For s ∈ S , let D ( s ) be thevirtual link diagram obtained from D by performing s ( p )-splice at each real crossing p of D . Define the weight of s ∈ S , denoted by h D / s i , by h D / s i : = A α ( s ) − β ( s ) ( − A − A − ) ♯ D ( s ) − ∈ Z [ A , A − ] , where α ( s ) : = ♯ s − ( A ), β ( s ) : = ♯ s − ( B ) and ♯ D ( s ) is the number of components of D ( s ).The Kau ff man bracket polynomial h D i of D is defined by h D i : = X s ∈S h D / s i . For a Laurent polynomial f ∈ Z [ A , A − ] we denote the maximal (resp. the minimal) degreeof f by deg f (resp. deg f ), and define span( f ) asspan( f ) : = deg f − deg f . It is well known that span h D i is invariant under the generalized Reidemeister moves. Fora virtual link L we define span( L ) as span h D i for any diagram D representing L . Thefollowing property is well known. Lemma 2.1. If L is a classical link, then span( L ) can be divided by 4.2.2. The Turaev surface.
Recall the definition of the Turaev surface of a virtual linkdiagram D with c ( D ) >
0. We replace all the real crossings of D with disks and jointhese disks with bands, each of which corresponds to an arc of D , so that the resultingsurface F ′ ( D ) is orientable and is almost embedded into R , except for the neighborhoodsof virtual crossings (Figure 2.2). The surface F ′ ( D ) with boundary has been introduced in[3] and is called the fat frame of D . Around the real crossings of D , we color ∂ F ′ ( D ) incheckerboard manner so that the components corresponding to curves obtained by A-splice(resp. B-splice) is colored by red (resp. blue), as in Figure 2.3. We twist each band of F ′ ( D )if necessary so that the checkerboard colorings of ∂ F ′ ( D ) near the real crossings extendconsistently to whole boundary, as in Figure 2.4. The resulting surface with boundary iscalled the twisted fat frame of D , and denoted by F ( D ). Now let T D be the closed surfaceobtained by attaching disks along ∂ F ( D ), and call T D the Turaev surface of D . Remark 2.2.
By definition the Turaev surface of a trivial link diagram is a union of S . Remark 2.3. T D is orientable for any classical link diagram [11]. This is not necessarilythe case for virtual diagrams; see Figure 2.4 for example. It is not hard to see that F ( D ) = F ′ ( D ), and hence T D is orientable, if D is alternating. A ABB F igure F ′ ( D ) around the real crossings of D disks F igure Definition 2.4.
For a virtual link diagram D , let s A (resp. s B ) be the state of D that mapsevery real crossing of D to A (resp. B). Remark 2.5.
By construction, the red boundary of ∂ F ( D ) coresspond to D ( s A ), and theblue boundary of ∂ F ( D ) coresspond to D ( s B ). Thus we have a one-to-one corespondencebetween the components of D ( s A ) ∪ D ( s B ) and those of ∂ F ( D ). Lemma 2.6.
For a virtual link diagram D , let χ ( D ) be the Euler characteristic of T D . Thenwe have ♯ D ( s A ) + ♯ D ( s B ) = χ ( D ) + c ( D ). Proof. T D can be decomposed into a cell complex; 0-cells are in one-one correspondenceto the real crossings of D , 1-cells correspond to the arcs of D , and 2-cells corresspond tothe disks attached to F ( D ) along the boundary. The number of arcs is equal to 2 c ( D ), since D can be seen as a 4-valent graph whose vertices are the real crossings of D and whoseedges are the arcs of D . Since the number of 2-cells is ♯ D ( s A ) + ♯ D ( s B ) as mentioned inRemark 2.5, we have χ ( D ) = c ( D ) − c ( D ) + ♯ D ( s A ) + ♯ D ( s B ) = − c ( D ) + ♯ D ( s A ) + ♯ D ( s B ) . (cid:3)
3. P seudo - adequate daigrams and v - pseudo - adequate diagrams Definition 3.1.
A virtual link diagram D is said to be pseudo-adequate if, for any s ∈ S ,(a) ♯ D ( s A ) ≥ ♯ D ( s A (1)) and(b) ♯ D ( s B ) ≥ ♯ D ( s B (1)),where s A (1) (resp. s B (1)) is a state that sends one real crossing to B (resp. A) and all theother crossings to A (resp. B); for notations see §4. Remark 3.2.
A virtual link diagram D is said to be A-adequate (resp.
B-adequate ) if thecondition (a) (resp. (b)) in Definition 3.1 holds after replacing “ ≥ ” with “ > ”, and adequate if D is A-adequate and B-adequate [7]. D is adequate if and only if the four componentsof ∂ F ( D ) near each real crossing p are all distinct. MINORI OKAMURA AND KEIICHI SAKAI · · · H n · · · H n ( s A ) · · · H n ( s A (1)) · · · H ′ n F igure H n is pseudo-adequate but not adequate, and H n is v-pseudo-adequate obtained from H ′ n (1) (2)F igure H n in Fig-ure 3.1 is pseudo-adequate but not adequate, because ♯ H n ( s A ) = ♯ H n ( s A (1)) = ♯ H n ( s B ) = ♯ H n ( s B (1)) = virtualization of a real crossing is the replacement of the real crossing with a virtualcrossing. Definition 3.3. A v-adequate (resp. v-pseudo-adequate ) diagram is a diagram D obtainedfrom an adequate (resp. a pseudo-adequate) diagram D ′ by virtualizing one real crossingof D ′ . Example 3.4.
The diagram H n in Figure 3.1 is v-pseudo-adequate; indeed H n can beobtained from H ′ n by virtualizing one of its real crossings, and H ′ n is pseudo-adequatebecause ♯ H ′ n ( s A ) = ♯ H ′ n ( s B ) = ♯ H ′ n ( s A (1)) and ♯ H ′ n ( s B (1)) are 1 or 2. Definition 3.5.
For s ∈ S , a connecting arc γ in D ( s ) is said to be admissible if(1) γ connect two distinct components of D ( s ), or(2) both the endpoint of γ are on a single component of D ( s ) and any orientation of thecomponent looks as in Figure 3.2.4. T he refined KMT inequality
Theorem 4.1 ([1], see also [2, 8]) . Let D be a virtual link diagram representing a virtuallink L , and χ ( D ) be the Euler characteristic of T D . Then we havespan( L ) ≤ c ( D ) + χ ( D ) − . It is known that the equality holds in Theorem 4.1 for adequate diagrams. This conditioncan be refined as follows:
Theorem 4.2.
The equality holds in Theorem 4.1 for pseudo-adequate diagrams.The above Theorem 4.2 deduces our main theorem.
Theorem 4.3.
Let D be a v-pseudo-adequate virtual link diagram obtained from a pseudo-adequate diagram D ′ by replacing a real crossing p with a virtual crossing v p . Let C A and C B be the components of D ( s A ) and D ( s B ) including v p respectively. If any connectingarcs of D ( s A ) (resp. D ( s B )) whose endpoints are both on C A (resp. C B ) are admissible, thenspan h D i = c ( D ) + χ ( D ) − Corollary 4.4.
Let D be a v-adequate diagram obtained from an adequate classical dia-gram D ′ . Then D does not represent any classical link. Proof.
Any connecting arcs of D ( s A ) and D ( s B ) whose endpoints are both on C A and C B (Theorem 4.3) are admissible, because both C A and C B contain exactly one virtual crossing;see Figure 3.2. Theorem 4.3 therefore implies span h D i = c ( D ) + χ ( D ) − D ′ is adequate (and hence pseudo-adequate), Theorem 4.2implies span h D ′ i = c ( D ′ ) + χ ( D ′ ) − c ( D ′ ) = c ( D ) + ♯ D ′ ( s A ) = ♯ D ( s A ) + ♯ D ′ ( s B ) = ♯ D ( s B ) +
1, we obtain χ ( D ′ ) = χ ( D ) + (cid:10) D ′ (cid:11) = c ( D ) − + χ ( D ) − − = c ( D ) + χ ( D ) − − = span h D i − . Since span h D ′ i is divisible by 4 by Lemma 2.1, span h D i cannot be divided by 4. Thus D cannot represent any classical link. (cid:3) Remark 4.5.
In his master’s thesis, Kishino [6] proved that the span h D i = c ( D ) − D is obtained from a proper alternating classical diagram, and hence D doesnot represent any classical link. Corollary 4.4 generalizes his result; any proper alternatingdiagram is adequate. Theorem 4.3 has another application to proper alternating virtualdiagrams; see [3]. Example 4.6.
The diagram H n in Figure 3.4 is pseudo-adequate and v-pseudo-adequate.Moreover the endpoints of any connecting arcs of H n ( s A ) (resp. K n ( s B )) are admissible.Thus both Theorems 4.2 and 4.3 can be applied to deduce span h H n i = c ( H n ) + χ ( H n ) − c ( H n ) = n and we can see that ♯ H n ( s A ) = ♯ H n ( s B ) =
1, Lemma 2.6 tells us that χ ( K n ) = − n and hence span h K n i = n . In particular, if n is odd, then span h H n i cannotbe divided by 4 and therefore H n cannot represent any classical links.The rest of this paper is devoted to the proofs of Theorems 4.2 and 4.3.For arbitrarily chosen real crossings p , . . . , p j (1 ≤ j ≤ c ( D )), let s A ( j ) (resp. s B ( j )) bea state of D that maps p , . . . , p j to B (resp. A) and the other real crossings to A (resp. B).Note that any s ∈ S other than s A and s B can be expressed as s A ( j ) or s B ( j ). By definitiondeg h D / s A ( j ) i = c ( D ) − j + ♯ D ( s A ( j )) − , (4.1) deg h D / s B ( j ) i = − c ( D ) + j − ♯ D ( s B ( j )) + . (4.2) Lemma 4.7. If j ≥
1, then we havedeg h D / s A ( j ) i ≤ deg h D / s A i , deg h D / s B ( j ) i ≥ deg h D / s B i . Thus span h D i ≤ deg h D / s A i − deg h D / s B i . Proof.
Figure 4.1 shows ♯ D ( s A ( j − − ≤ ♯ D ( s A ( j )) ≤ ♯ D ( s A ( j − + . (4.3)By (4.3) we inductively deduce(4.4) ♯ D ( s A ( j )) ≤ ♯ D ( s A ( j − + ≤ ♯ D ( s A ( j − + ≤ · · · ≤ ♯ D ( s A ) + j . MINORI OKAMURA AND KEIICHI SAKAI
A-splice B-splice D ( s A ( j − D ( s A ( j − D ( s A ( j − D ( s A ( j )) D ( s A ( j )) D ( s A ( j )) ppp A-splice B-spliceA-splice B-spliceF igure ♯ D ( s B ( j )) ≤ ♯ D ( s B ) + j . (4.5)By definition deg h D / s A i = c ( D ) + ♯ D ( s A ) − , (4.6) deg h D / s B i = − c ( D ) − ♯ D ( s B ) + . (4.7)By (4.4)-(4.7), deg h D / s A ( j ) i = c ( D ) − j + ♯ D ( s A ( j )) − ≤ c ( D ) + ♯ D ( s A ) − , deg h D / s B ( j ) i = − c ( D ) + j − ♯ D ( s B ( j )) + ≥ − c ( D ) − ♯ D ( s B ) + . These inequalities and (4.6), (4.7) implydeg h D i ≤ deg h D / s A i and deg h D i ≥ deg h D / s B i . Thus we have span h D i ≤ deg h D / s A i − deg h D / s B i . (cid:3) Proof of Theorem 4.1.
Lemma 2.6 and Lemma 4.7 implyspan h D i ≤ deg h D / s A i − deg h D / s B i (4.8) = c ( D ) + ♯ D ( s A ) + ♯ D ( s B )) − = c ( D ) + χ ( D ) + c ( D )) − = c ( D ) + χ ( D ) − . (cid:3) Proof of Theorem 4.2.
Pseudo-adequacy of D implies(4.9) ♯ D ( s A (1)) ≤ ♯ D ( s A ) , ♯ D ( s B (1)) ≤ ♯ D ( s B ) . Thus (4.4) and (4.5) can be sharpend in this case as ♯ D ( s A ( j )) ≤ ♯ D ( s A ) + j − , ♯ D ( s B ( j )) ≤ ♯ D ( s B ) + j − . (4.10)(4.6), (4.7) and (4.10) implydeg h D / s A ( j ) i = c ( D ) − j + ♯ D ( s A ( j )) − ≤ c ( D ) + ♯ D ( s A ) − , (4.11) deg h D / s B ( j ) i = − c ( D ) + j − ♯ D ( s B ( j )) + ≥ − c ( D ) − ♯ D ( s B ) + . (4.12)These estimations deduce deg h D / s A ( j ) i < deg h D i and deg h D / s B ( j ) i > deg h D i for anychoices of p , . . . , p j ( j ≥ h D i = deg h D / s A i and deg h D i = deg h D / s B i . Thus the inequality in (4.8) becomes an equality. (cid:3) Proof of Theorem 4.3.
In general, it is not hard to see that a connecting arc in D ( s ) ( s ∈ S )corresponing to a real crossing p is admissible if and only if ♯ D ( s (1)) ≤ ♯ D ( s ) where s (1)is the state of D obtained from s by changing splice at p . Thus if all the connecting arcs of D ( s A ) and those of D ( s B ) are admissible, then D is pseudo-adequate.If D is such a diagram as in Theorem 4.3, all the connecting arcs of D ( s A ) and D ( s B ) areadmissible; because D ′ is (pseudo-)adequate, the connecting arcs one of whose endpointsis not on C A nor C B are admissible. By assumption the other arcs are also admissible. (cid:3) A cknowledgements The authors express their sincere appreciation to Professor Naoko Kamada and Profes-sor Shin Satoh for their fruitful comments and encouragements. KS is partially supportedby JSPS KAKENHI Grant Number 16K05144.R eferences [1] Y. Bae, H. R. Morton,
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