aa r X i v : . [ m a t h . G T ] D ec ON THE QUESTION OF GENERICITYOF HYPERBOLIC KNOTS
ANDREI MALYUTIN
Abstract.
A well-known conjecture in knot theory says that the percentageof hyperbolic knots amongst all of the prime knots of n or fewer crossingsapproaches as n approaches infinity. In this paper, it is proved thatthis conjecture contradicts several other plausible conjectures, including the120-year-old conjecture on additivity of the crossing number of knots underconnected sum and the conjecture that the crossing number of a satellite knotis not less than that of its companion. Introduction
William Thurston proved in 1978 that every non-torus non-satellite knot is ahyperbolic knot. Computations show that the overwhelming majority of primeknots with small crossing number are hyperbolic knots. The following tablegives the number of hyperbolic, prime satellite, and torus knots of n crossingsfor n = 3 , . . . , (see [HTW98] or the sequences A002863, A052408, A051765,and A051764 in the Sloane’s encyclopedia of integer sequences). type \ n = Table 1.
Number of prime knots(A part of) these data gave rise to the following conjecture (see [Ad94b, p. 119]).
Conjecture 1.
The percentage of hyperbolic knots amongst all of the prime knotsof n or fewer crossings approaches as n approaches infinity. In the present paper, we show that Conjecture 1 contradicts several other longstanding conjectures, including the following one.
Conjecture 2.
The crossing number of knots is additive with respect to connectedsum.
See, e. g., [Ad94b, p. 69], [Kir97, Problem 1.65], and [La09] for comments andrelated results. Another related conjecture is as follows.
Conjecture 3.
The crossing number of a satellite knot is bigger (a weaker variant:not less) than that of its companion.
See [Ad94b, p. 118], [Kir97, Problem 1.67 (attributed to de Souza)], and [La14].It is remarked in [Kir97, Problem 1.67] concerning Conjecture 3 that ‘Surely theanswer is yes, so the problem indicates the difficulties of proving statements about the crossing number’. Since a composite knot is a connected sum of its factorsand, at the same time, is a satellite of each of its factors, the ‘intersection’ ofConjectures 2 and 3 yields the following.
Conjecture 4.
The crossing number of a composite knot is bigger (a weakervariant: not less) than that of each of its factors.
Let us denote by cr( X ) the crossing number of a knot X . If P is a prime knotand λ is a real number, we say that P is λ -regular if we have cr( K ) ≥ λ · cr( P ) whenever P is a factor of a knot K . In this terminology, Conjecture 4 says thateach prime knot is -regular. Lackenby [La09] proved that each knot is -regular.Our considerations involve the following conjecture. Conjecture 5.
Each prime knot is -regular. We also consider the following weakening of Conjecture 5.
Conjecture 6.
There exist ε > and N > such that, for all n > N , thepercentage of -regular knots amongst all of the hyperbolic knots of n or fewercrossings is at least ε . We have the following obvious implications.Conj. 4 = ⇒ Conj. 5 = ⇒ Conj. 6Conj. 2 = ⇒ Conj. 3 = ⇒ The main result of this paper is the following theorem.
Theorem 1.
Conjecture 1 contradicts (each of ) Conjectures 2, 3, 4, 5, and 6.
The paper is organised as follows. Section 2 contains remarks concerning Con-jectures 1–6. In Section 3, we present the key idea of the proof of Theorem 1 andreduce Theorem 1 to Proposition 1 consisting of three assertions. Sections 4–8contain the proof of Proposition 1. In Sections 4 and 5, we prove the first twoassertions of Proposition 1. Section 6 contains a combinatorial lemma used in theproof of the last assertion of Proposition 1. Section 7 contains preliminaries ontangles. In Section 8, we prove the last assertion of Proposition 1. In Section 9,we introduce a new property of knots (strong property PT) and prove Theorem 3strengthening Theorem 1. In Section 10, we show that an assumption that Con-jectures 2–5 has many strong counterexamples contradicts Conjecture 1 as well.In Section 11, we show that certain assumptions concerning unknotting numbersof knots contradict Conjecture 1.The paper should be interpreted as being in either the PL or smooth category.For standard definitions we mostly use the conventions of [BZ06] and [BZH14].There will be a certain abuse of language in order to avoid complicating thenotation. In particular, a knot K will be a circle embedded in a -sphere S , apair ( S , K ) , or a class of homeomorhic pairs (cf. [BZ06, p. 1]). No orientationson knots and spaces are placed if not otherwise stated.The author is grateful to Ivan Dynnikov, Evgeny Fominykh, Aleksandr Gai-fullin, Vadim Kaimanovich, Maksim Karev, Paul Kirk, Vladimir Nezhinskij, SemënPodkorytov, Józef Przytycki, Alexey Sleptsov, and Andrei Vesnin for helpful com-ments and suggestions. N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 3 Remarks
We list certain results related to Conjectures 1–6.
Predominance of hyperbolic objects.
In recent years, a number of resultshave been obtained showing the predominance of hyperbolic objects in variouscases. We refer to the works of Ma [Ma14] and Ito [Ito15, Theorem 2] for re-sults concerning genericity of hyperbolic knots and links. See also [Mah10a],[LMW14, Theorem 2], [LusMo12], [Riv14], and [Ito15, Theorem 1] for results ongenericity of hyperbolic 3-manifolds. Related results show that pseudo-Anosovsprevail (in various senses) in mapping class groups of surfaces. We refer to[Riv08, Riv09, Riv10, Riv12, Riv14], [Kow08], [Mah10b, Mah11, Mah12], [AK10],[Sis11], [Mal12], [LubMe12], [MS13], and also [Car13, CW13, Wi14] (non-randomapproach) for precise statements and detailed discussions. See also [GTT16] forthe genericity of loxodromic isometries for actions of hyperbolic groups on hy-perbolic spaces. Other examples of hyperbolicity predominance can be found inextensive literature on exceptional Dehn fillings (see [Thu79] etc.) and in [Gro87,0.2.A], [GhH90, p. 20], [Ch91] and [Ch95], [Ols92], [Gro93], [Żuk03], [Oll04],[Oll05], where viewpoints are given from which it appears that a generic finitelypresented group is word hyperbolic. Apparently, combining approaches devel-oped by [Ito15] and [Ma14] with results of [Car13, CW13, Wi14] (the same for[LusMo12]) one can obtain more viewpoints where generic knot will be hyperbolic.
Predominance of non-hyperbolic objects.
As for natural models where it isproved that hyperbolic objects are rare, we have the standard methods of gen-erating knots as polygons in R . Under this approach, composite knots prevailand prime knots (including hyperbolic ones) are asymptotically scarce. See Sum-ners and Whittington [SW88], Pippenger [Pip89], and also Soteros, Sumners, andWhittington [SSW92] for the case of self-avoiding random polygons on the simplecubic lattice; see [Oetal94] and [Sot98] for such polygons in specific subsets of thelattice; see [DPS94] and [Jun94] for local and global knotting in Gaussian randompolygons; see [Di95] and [DNS01] for local and global knotting in equilateral ran-dom polygons; see also [Ken79] for knotting of Brownian motion and [Sum09] and[MMO11] for more references. An interesting idea has appeared in [Ad05, p. 4] and[Cr04, p. 95] that prime satellite knots should prevail over hyperbolic ones whenwe consider Gaussian random polygons. In both [Ad05] and [Cr04], however, theidea was apparently inspired by a misinterpretation of results in [Jun94]. Crossing number additivity.
Murasugi [Mur87, Corollary 6] proved that Con-jecture 2 is valid for alternating knots. (This follows from the proof of Tait con-jecture that reduced alternating projections are minimal; this Tait conjecture wasalso proved, independently, by Kauffman [Kau87] and Thistlethwaite [Thi87].)Conjecture 2 is valid for adequate knots (see [LT88]). Diao [Di04] and Gru-ber [Gru03] independently proved that Conjecture 2 is valid for torus knots andcertain other particular classes of knots. Results of [Mur87], [Kau87], [Thi87] im-ply that alternating knots are 1-regular. Diao [Di04, Theorem 3.8] showed that It is shown in [Mur87], [Kau87], [Thi87] that (i) for each knot K we have span V K ( t ) ≤ cr( K ) ,and (ii) for alternating K we have span V K ( t ) = cr( K ) , where V K ( t ) is the Jones polynomialof K and span V K ( t ) denotes the difference between the maximal and minimal degrees of V K ( t ) .(It is known that V K ( t ) = 0 so that span V K ( t ) is well-defined. See [Jon85, Theorem 15].) Then1-regularity of alternating knots follows because, for any knots K and K , we have [Jon85, ANDREI MALYUTIN torus knots are 1-regular. In [PZ15], the authors introduce a telescopic familyof conjectures concerning monotonic simplification of link diagrams and providesupporting evidence for (the strongest of) these conjectures. Each of Petronio–Zanellati conjectures implies Conjecture 2.
Torus knots.
Murasugi [Mur91, Proposition 7.5] proved that the torus link oftype ( p, q ) , where ≤ p ≤ q , has crossing number ( p − q . Taking into ac-count that the number of all prime knots of n crossings grows exponentially in n (see [ES87, Wel92]), this implies that the percentage of torus knots amongst allof the prime knots of n or fewer crossings approaches as n approaches infinity.Thus, only satellite knots pose a danger to Conjecture 1. Hyperbolic knots.
Several interesting classes of knots are known to consist ofhyperbolic and torus knots only. In particular, amongst these classes are:– prime alternating knots, including 2-bridge knots (see [Men84]),– prime almost alternating knots (see [Aetal92]),– prime toroidally alternating knots (see [Ad94a]),– arborescent knots, including 2-bridge knots, pretzel knots, and Montesinosknots (see [BS10], Theorem 1.5 and subsequent discussion in [FG09]), etc.More families of hyperbolic knots, links, and tangles are listed in [Ad05]. See[Ito11], [IK12, Theorems 8.3, 8.4] for new examples of huge classes of hyberbolicknots, links, and -manifolds.3. The idea of the proof of Theorem 1
Our proof of Theorem 1 uses a specific way of constructing satellite knots. Forbrevity, we use the term γ -knots for the satellite knots constructed in this way. Definition. γ -Knots. Let K be a knot in a -sphere S , and let V be an un-knotted solid torus in S such that K is contained in the interior of V . Let ψ : V → W ⊂ S be a homeomorphism onto a tubular neighbourhood W of ahyperbolic knot. Recall that the winding number of K in V is the absolute valueof the algebraic intersection number of K with a meridional disk in V . Assumethat the winding number of K in V is at least and that ψ maps a longitude of V to a longitude of W . Then we say that the knot ψ ( K ) ⊂ S is a γ -knotover K .A method of constructing a γ -knot is given in Fig. 1. Assume that a diagram D ′ of a knot K ′ is obtained from a diagram D of a knot K by local move as in Fig. 1.(See Fig. 2 for an example.) Our definitions imply that if two arrows on arcs inFig. 1(a) indicate the same orientation on K , then K ′ is a γ -knot over K . Here,the winding number is while the companion hyperbolic knot is the figure-eightknot. (In order to check that the condition on longitudes is also fulfilled, weobserve that each arc in Fig. 1 has zero total curvature.) Theorem 6] span V K ♯K ( t ) = span V K ( t ) + span V K ( t ) . Ivan Dynnikov (private communication) found a counterexample to Petronio–Zanellaticonjectures. If a solid torus U is embedded in a -sphere S , then there exists an essential curve in ∂U that bounds a -sided surface in S \ int( U ) (a Seifert surface). This curve is unique up toisotopy on ∂U and is called a longitude of U in S (see, e. g., [BZ06, Theorem 3.1]). N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 5 (a) (b)
Figure 1.
Double figure-eight moveWe deduce Theorem 1 from the following proposition on γ -knots. Proposition 1. (i)
Each γ -knot is a satellite knot. (ii) The sets of γ -knots over distinct non-satellite knots are disjoint. (iii) If P is a -regular prime knot, then there exists a prime γ -knot P ′ over P with cr( P ′ ) ≤ cr( P ) + 17 . Remark.
Assertion (iii) of Proposition 1 is not obvious because a γ -knot over aprime knot is not necessarily prime (see Fig. 2). Figure 2.
A composite γ -knot over the trefoil Proposition implies Theorem . We introduce the following notation. Let p n (resp., h n , s n ) denote the number of prime (resp., hyperbolic, prime satellite)knots with crossing number n . We set P n = P nk =1 p n , H n = P nk =1 h n , and S n = P nk =1 s n .Since each of Conjectures 2, 3, 4, and 5 implies Conjecture 6 (see the diagrambefore Theorem 1), it suffices to prove only that Conjectures 6 and 1 are incom-patible. If Conjecture 6 is true, then there exist ε > and N > such that, forall n > N , the number of -regular hyperbolic knots of n or fewer crossings is atleast ε H n . Obviously, in this case assertions (i), (ii), and (iii) of Proposition 1imply that (for all n > N ) we have S n +17 ≥ ε H n . ANDREI MALYUTIN
Therefore, we have P n +17 ≥ H n +17 + ε H n . This is equivalent to the following inequality(1) ≥ H n +17 P n +17 + ε H n P n P n P n +17 . If Conjecture 1 is true, then both sequences H n +17 P n +17 and H n P n tend to . In this case,Eq. (1) implies that P n +17 P n n → + ∞ −−−−−→ + ∞ . Consequently, for each
B > we have P n > B n for all sufficiently large n . (Weconsider subsequences of the form P n +17 i , i ∈ N .) In other words, we have(2) P /nn n → + ∞ −−−−−→ + ∞ . However, it is shown in [Wel92] that lim sup k →∞ p /nn < + ∞ , which implies that there exists B > such that p n < B n for all n ∈ N . Then, foreach n ∈ N we have P n < ( B + 1) n whence it follows that lim sup k →∞ P /nn ≤ B + 1 < + ∞ . This contradicts (2). The obtained contradiction completes the proof. (cid:3) Proof of assertion (i) of Proposition 1
We recall definitions of satellite knots. A knot K in S is a satellite knot if S contains a non-trivial knot C such that K lies in the interior of a regularneighbourhood V of C , V does not contain a -ball containing K , and K is nota core curve of the solid torus V . The knot K is a satellite knot if and only if K contains an incompressible, non-boundary parallel torus in its complement. (Fora proof, see [BZH14, Remark 16.1, p. 335].)Let K be a γ -knot in S . Then the definition of γ -knots implies that K liesin a knotted solid torus W ⊂ S such that the winding number of K in W is atleast . Since the winding number of K in W is at least , it follows that W doesnot contain a -ball containing K , and K is not a core curve of V . This meansby the above definition that K is a satellite knot.5. Proof of assertion (ii) of Proposition 1
We show that the sets of γ -knots over distinct non-satellite knots are disjoint.Suppose to the contrary that there exist a knot K and two distinct non-satelliteknots H and H such that K is a γ -knot both over H and over H . By thedefinition of γ -knots, this means that there exist embedded solid tori V and V in S and re-embeddings φ : V → S and φ : V → S such that, for each i ∈ { , } , the following conditions hold:– V i is a tubular neighbourhood of a hyperbolic knot,– K lies in the interior of V i and the winding number of K in V i is at least ,– the solid torus φ i ( V i ) is unknotted,– φ i maps a longitude of V i to a longitude of φ i ( V i ) , N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 7 – we have φ i ( K ) = H i . Claim 1.
The tori ∂V and ∂V are both incompressible in S \ K . Since the winding number of K in V i is non-zero, it follows that no -ball in V i contains K . If a knotted solid torus U in a -sphere S contains a knot L in itsinterior while no -ball in U contains L , then ∂U is sometimes called a companiontorus of L . It is well known that, in this case, ∂U is incompressible in S \ L .(See, e. g., [BZH14, Propositions 3.10 and 3.12, and E 2.9].) This implies Claim 1. Claim 2.
There exists an isotopy of ∂V in S \ K that moves ∂V to a positionwhere ∂V ∩ ∂V = ∅ . It may be assumed that ∂V intersects ∂V transversely in simple closed curves.If the intersection ∂V ∩ ∂V contains a curve that is inessential in ∂V , let C bean innermost of such curves and let d be the open disk in ∂V \ ∂V bounded by C .Then C is inessential in ∂V because ∂V is incompressible in S \ K (Claim 1). Let δ be the open disk in ∂V bounded by C ( δ may intersect ∂V ). Then the sphere d ∪ δ ∪ C bounds a ball (say, B ) in S \ K . We have B ∩ ∂V = δ ∪ C . It followsthat we can eliminate C (together with δ ∩ ∂V , if nonempty) by an isotopy of ∂V in a neighborhood of B . Therefore, we can eliminate all components of ∂V ∩ ∂V that are inessential in ∂V . The remaining curves of ∂V ∩ ∂V are essential in ∂V as well. (For if C is an innermost of inessential curves from ∂V ∩ ∂V on ∂V ,then C is inessential in ∂V because ∂V is incompressible in S \ K by Claim 1.)Now, if ∂V ∩ ∂V is still nonempty, the space ∂V \ ∂V is a collection of annuli.It is known that every incompressible properly embedded annulus in the closureof the complement of a hyperbolic knot is boundary parallel (see, e. g., [BZ06,Lemma 15.26]). Applying this to the space S \ int( V ) , we see that there existsan isotopy of ∂V in S \ K moving ∂V in S \ ∂V . Claim 2 is proved.The classical Isotopy Extension Theorem (for smooth manifolds) says that if A is a compact submanifold of a manifold M and F : A × I → M is an isotopy of A with F ( A × I ) ⊂ int( M ) , then F extends to an ambient isotopy (i. e., a diffeotopyof M ) having compact support (see, e. g., [Hir76, p. 179]). Applying this theoremto the isotopy of ∂V in S \ K from Claim 2 yields the following. Claim 3.
There exists an ambient isotopy of S , fixing K pointwise, that moves V to a position in which ∂V ∩ ∂V = ∅ . Thus, we can assume without loss of generality that ∂V ∩ ∂V = ∅ (while V and V satisfy all properties listed at the beginning of the proof). Now, let M and M denote the closures of the complements S \ V and S \ V respectively. Claim 4. M and M are disjoint. In order to prove Claim 4, we need the following assertion.
Claim 5.
There is no isotopy between ∂V and ∂V in S \ K . Suppose to the contrary that such an isotopy exists. Then the Isotopy Ex-tension Theorem (see above) implies that there exists an ambient isotopy of S ,fixing K pointwise, that moves ∂V to ∂V . This yields an isotopy between V and V that fixes K pointwise. Then the triples ( V , K, ℓ ) and ( V , K, ℓ ) , where ℓ i is a longitude of V i , i = 1 , , are homeomorphic, i. e., there exists a homeo-morphism τ : V → V such that τ ( K ) = K and τ ( ℓ ) = ℓ . This implies that ANDREI MALYUTIN the pairs ( S , φ ( K )) and ( S , φ ( K )) are homeomorphic. Indeed, we observethat ( S , φ i ( K )) is obtained from ( V i , K ) by a Dehn filling along ℓ i , that is, ( S , φ i ( K )) is obtained by attaching a solid torus V to V i by a gluing homeomor-phism σ i : ∂V → ∂V i such that σ − i ( ℓ i ) bounds a meridional disk of V . Thus, thehomeomorphism τ : V → V extends to a homeomorphism S → S that maps φ ( K ) to φ ( K ) . This means that the knots H and H are equivalent becausewe have φ i ( K ) = H i by construction. This contradicts the assumption that H and H are distinct. Claim 5 is proved.Now, we pass to the proof of Claim 4. Observe that neither M contains ∂M = ∂V nor M contains ∂M = ∂V because an incompressible torus in a hyperbolicknot complement is boundary parallel by Thurston’s hyperbolization theorem,while ∂M = ∂V and ∂M = ∂V are not parallel by Claim 5. Obviously, thisimplies that M and M are disjoint.Another fact that we need is implied by the following proposition. Proposition 2.
Let C , C , . . . , C n be n disjoint submanifolds of S such thatfor all i ∈ { , , . . . , n } , K i = clos( S \ C i ) is a non-trivially embedded solid-torusin S . Then there exists n disjointly embedded 3-balls B , B , . . . , B n ⊂ S suchthat C i ⊂ B i for all i ∈ { , , . . . , n } . Moreover, each B i can be chosen to be C i union a -handle which is a tubular neighbourhood of a meridional disk for K i .Proof. See [Bud06, Proposition 2.1] and references therein for earlier proofs. (cid:3)
Applying Proposition 2 to M and M , we obtain the following claim. Claim 6.
There exists a meridional disk D for V such that D ⊂ V . Now, since we have M ⊂ V (Claim 4), the image φ ( M ) is well defined. Weconsider the complement W := S \ φ (int( M )) . Due to Alexander’s theoremon embedded torus in S , we observe that W is a knotted solid torus because weknow that the boundary ∂W = ∂φ ( M ) = φ ( ∂M ) = φ ( ∂V ) is a torus, whilethe complement S \ W = φ (int( M )) is homeomorphic to int( M ) , which is thecomplement of the knotted solid torus V . (Of course, by the Gordon–Luecketheorem we know, moreover, that W is a tubular neighbourhood of a hyperbolicknot.) We see that W contains φ ( K ) by construction. Finally, we see thatthe winding number of φ ( K ) in W is equal to the winding number of K in V because there exists a meridional disk D for V such that D ⊂ V so that φ maps D to a meridional disk of W . Therefore, φ ( K ) is contained in a knottedsolid torus W and the winding number of φ ( K ) in W is at least . This meansthat φ ( K ) is a satellite knot. Since we have H = φ ( K ) , this contradicts theassumption that H is not a satellite knot. This contradiction completes the proofof assertion (ii) of Proposition 1.6. A combinatorial lemma
The present section contains a lemma which is used in the proof of assertion(iii) of Proposition 1.
Definitions.
Let K be a knot in the -sphere S = R ∪ {∞} , and let D ⊂ S be a projection of K on a -sphere S = R ∪ {∞} in S . A knot projection issaid to be regular if its only singularities are transversal double points. If D isa regular knot projection, an edge in D is the closure of a component of the set N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 9 D \ V , where V is the set of double points of D . We say that two edges I and J of D are neighboring edges or neighbors if there exists a component Q of S \ D such that the boundary ∂Q contains both I and J . We say that two edges I and J of D are consecutive if the union I ∪ J is the image of a (connected) arc of theknot. We will denote by ρ the maximal metric on the set E ( D ) of edges of D inthe class of metrics satisfying the condition ρ ( I, J ) = 1 if I and J are consecutive edges of D . Lemma 1.
Any regular knot projection with n > double points has a pair ofneighboring edges I and J with ρ ( I, J ) ≥ n/ .Proof. Let D ⊂ S be a regular knot projection with n double points. We considerthe case with n ≥ (the case n = 1 is obvious). Observe that D has n edges.Put k := ⌊ n/ ⌋ , the largest integer not greater than n/ , and split the set E ( D ) of edges of D in three parts, E , E , and E , such that each part is a chain ofconsecutive edges, two parts consist of k edges each, and the third part consistsof n − k edges. (Note that n − k ∈ { k, k + 1 , k + 2 } ; in particular, we have n − k ≥ k , that is, each part consists of at least k edges. No part is emptysince we assume n ≥ .) Let D i ⊂ D , i = 1 , , , be the union of edges from E i .Observe that each D i is compact and connected and D = D ∪ D ∪ D . Let ussmoothly embed S in R as a sphere of radius and let dist denote the metricon S induced by the euclidean metric in R . For each i ∈ { , , } we set R i := { x ∈ S : dist( x, D i ) = dist( x, D ) } . Observe that R ∪ R ∪ R = S because D = D ∪ D ∪ D . We see that for each i ∈ { , , } the set R i is closed because D i is compact (consider a convergentsequence of points in R i ). Also, we see that for each i ∈ { , , } the set R i isconnected. Indeed, if p ∈ R i , then due to compactness of D i there exists a point q ∈ D i such that dist( p, q ) = dist( p, D ) . Then the geodesic segment between p and q is in R i by the triangle inequality. Therefore, R i is connected because D i isconnected. Finally, we see that for any { i, j } ⊂ { , , } the intersection R i ∩ R j is not empty because D i ⊂ R i and D j ⊂ R j , while D i ∩ D j is not empty.Thus, the sets R , R , and R satisfy assumptions of Lemma 2 below. Lemma 2implies that R , R , and R have a common point x . Clearly, x is not an innerpoint of an edge of D , so we have two possible cases:1) x is a double point of D ,2) x / ∈ D .Suppose x is a double point of D . Then there exists a triple { J , J , J } ofedges of D incident to x such that J i ∈ E i for all i ∈ { , , } . Without loss ofgenerality we can and will assume that J and J are consecutive. Then J and J are neighbors, and J and J are neighbors. It is easily seen that we have ρ ( J , J ) ≥ k and if ρ ( J , J ) = k then ρ ( J , J ) = k + 1 , and the theorem follows.Suppose x ∈ S \ D . Let Q be the component of S \ D containing x . Observethat the set { y ∈ D : dist( x, y ) = dist( x, D ) } is contained in ∂Q ⊂ D and contains no double points of D (due to smoothnessof embedding S → R ). Therefore, since x ∈ R ∩ R ∩ R , for each i ∈ { , , } the set ∂Q ∩ D i contains at least one edge of D . This means that there exists atriple { J , J , J } of pairwise neighboring edges of D such that we have J i ∈ E i for all i ∈ { , , } . It is an easy exercise to check that this triple contains a pair { I, J } with ρ ( I, J ) ≥ ⌈ n/ ⌉ ≥ n/ . (cid:3) Lemma 2.
If a triple of pairwise intersecting closed connected sets cover a simplyconnected space, then these three sets have a common point.Proof.
This follows, e. g., from Theorem 5 of [Bog02] in the case m = 1 . (cid:3) Tangles
Our proof of assertion (iii) of Proposition 1 uses tangles. The present sectioncontains some preliminaries on tangles.
Definitions. A k -string tangle , where k ∈ N , is a pair ( B, t ) where B is a -balland t is the union of k disjoint arcs in B with t ∩ ∂B = ∂t . We mostly interestedin the cases where k ∈ { , } . Two tangles, ( B, t ) and ( A, s ) , are equivalent ifthere is a homeomorphism of pairs from ( B, t ) to ( A, s ) . A tangle ( B, t ) is trivial if B contains a properly embedded disk containing t . A tangle ( B, t ) is locallyknotted if B contains a ball B ′ such that ( B ′ , B ′ ∩ t ) is a nontrivial -string tangle.A -string tangle ( B, t ) is prime if it is neither locally knotted nor trivial. If ( B, t ) and ( A, s ) are k -string tangles and f : ( ∂B, ∂t ) → ( ∂A, ∂s ) is a homeomorphism,a link in S can be obtained by identifying the boundaries of the tangles using f .The result, ( B, t ) ∪ f ( A, s ) , is referred to as a sum of the two tangles. If ( B, t ) is a -string tangle and ( A, s ) is the trivial -string tangle, then there is a unique (upto a homeomorphism of pairs) knot which is a sum of ( B, t ) and ( A, s ) . This knotis called the closure of ( B, t ) . We say that a -string tangle ( B, t ) is a cable tangleif there exists an embedding f : I × I → B such that f ( I × I ) ∩ ∂B = I × ∂I and t = f ( ∂I × I ) , where I := [0 , . (We treat the trivial -string tangle as a cabletangle.) Clearly, each tangle ( B, t ) can be embedded in R in such a way that B becomes a Euclidean ball while the endpoints ∂t lie on a great circle of thisball and t is in general position with respect to the projection onto the flat discbounded by the great circle. The projection, with additional information of over-and undercrossings, then gives us a tangle diagram . Examples of tangle diagramsare given in Figs. 1, 2, and 4. Theorem 2 ([Lick81, Theorem 1]) . A sum of two -string prime tangles is aprime link. Lemma 3.
Each nontrivial cable -string tangle is prime.Proof. (See [Lick81, Examples (a) and (b)].) It is enough to observe that we can,in an obvious manner, add the trivial -string tangle to any cable -string tangle soas to create the trivial knot, which proves that the initial tangle has no local knots(this follows by the Unique Factorization Theorem by Schubert [Schu49]). (cid:3) Lemma 4.
No composite knot is a sum of a nontrivial cable -string tangle withthe trivial -string tangle.Proof. Suppose that a knot K in S is presented as a sum ( S , K ) = ( B, t ) ∪ f ( A, s ) , f : ( ∂A, ∂s ) → ( ∂B, ∂t ) , We use notation ⌈ n/ ⌉ for the smallest integer not less than n/ . N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 11 of a nontrivial cable -string tangle ( B, t ) with a trivial -string tangle ( A, s ) . Let f : ( ∂A, ∂s ) → ( ∂B, ∂t ) yields an obvious ‘trivializing’ sum for ( B, t ) , that is, the sum ( B, t ) ∪ f ( A, s ) is the trivial knot. (See left side of Fig. 3.) Let M denote the double coverof the -sphere B ∪ f A branched over the trivial knot t ∪ f s , and let M bethe double cover of the -sphere B ∪ f A branched over the knot t ∪ f s = K .Then M is homeomorphic to the -sphere, while M and M are related by aDehn surgery along the solid torus covering ( A, s ) . We observe that the solidtorus V A ⊂ M = S that covers ( A, s ) is knotted as a composite knot. Indeed,the definition of cable tangles imply that there is an obvious ambient isotopy of B ∪ f A that moves t ∪ f s and A to a position in which t ∪ f s is a geometriccircle and A is a closed regular neigborhood of a ‘knotted diameter’ of this circle.See Fig. 3. A A A
Figure 3.
For the proof of Lemma 4This clearly implies that V A is a regular neigborhood of a composite knot. (Thiscomposite knot is a sum of two copies of the -string tangle ( B, t ) , where t is acomponent of t .) It is known that a nontrivial Dehn surgery on a composite knotin S yields an irreducible (hence prime) manifold (see [Gor83, Theorem 7.1]). Itis known that if the double cover of S branched over a knot R is prime then R is prime (see [Wal69]; see also [KT80, Corollary 4] for the inverse implication).Consequently, K is a prime knot if nontrivial. (cid:3) Remarks.
1. Lemma 4 also follows from results of [E-M86] (see also [E-M88,Theorem 6]) or equivalently from the fact that only integral Dehn surgeries canyield reducible manifolds [GL87]. This way of proof uses the fact that cable knotsare prime (see [Schu53, p. 250, Satz 4], [Gra91, Cor. 2]).2. Lemma 4 is used in the proof of Proposition 4 (which in its turn is used in theproof of assertion (iii) of Proposition 1), where it covers the case of -bridge knots.It is known (see [Wel92]) that the percentage of -bridge knots amongst all of theprime knots of n or fewer crossings approaches as n approaches infinity. Thus,in the proof of Theorem 1, we can discard -bridge knots together with Lemma 4. In fact, the results of [BS86, BS88] imply that there is essentially unique way to create thetrivial knot as a sum of a given prime -string tangle and a trivial -string tangle. In particular,if φ : ( ∂A, ∂s ) → ( ∂B, ∂t ) is a homeomorphism such that ( B, t ) ∪ φ ( A, s ) is the trivial knot thenthe map f − ◦ φ : ( ∂A, ∂s ) → ( ∂A, ∂s ) extends to a map F : ( A, s ) → ( A, s ) such that F ( s ) = s . Nevertheless, we use Lemma 4 for the sake of completeness of Propositions 4and 1.
Corollary 1.
No composite knot is a sum of two cable -string tangles.Proof. A cable -string tangle is either prime or trivial (Lemma 3). A sum of twoprime -string tangles is a prime link by Theorem 2. No composite knot is a sumof a nontrivial cable -string tangle with the trivial -string tangle by Lemma 4.If a knot K is a sum of two trivial -string tangles, then the bridge number b( K ) of K is at most . If b( K ) = 1 then K is a trivial knot. If b( K ) = 2 then K aprime knot by [Schu54]. (cid:3) Proof of assertion (iii) of Proposition 1
Definition. Weak property PT.
Let D be a knot diagram on the -sphere S = R ∪ {∞} . We say that D has weak property PT (PT stands for ‘tangleprimeness’) if D is obtained by adding ears to a diagram of a tangle that is notlocally knotted (that is, the tangle is either prime or trivial). In other words, D has weak property PT if there exists a -disk d ⊂ S such that– the boundary ∂d intersects D transversely in four points;– the intersection d ∩ D consists of two simple non-intersecting arcs (as on theleft side of Fig. 1);– the complementary disk δ := S \ int( d ) with the diagram δ ∩ D represents a -string tangle ( B, t ) which is not locally knotted (that is, ( B, t ) is either primeor trivial).We say that a knot has weak property PT if it has a minimal diagram withweak property PT. Proposition 3.
Each minimal diagram of each -regular prime knot has weakproperty PT. In particular, each -regular prime knot has weak property PT.Proof. Let D P be a minimal diagram, on the -sphere S = R ∪ {∞} , of a -regular prime knot P . By Lemma 1, D P has a pair of neighboring edges I and J with ρ ( I, J ) ≥ P )3 . Since I and J are neighbors, there exists a disk d ⊂ S such that the intersection d ∩ D P consists of a subarc of I and a subarc of J , while ∂d intersects D P transversely in four points. Let δ denote the disk S \ int( d ) , andlet ( B, t ) be the -string tangle represented by the diagram δ ∩ D P . Let t and t be the components of t , and let K and K be the knots that are the closures ofthe -string tangles ( B, t ) and ( B, t ) . Claim 7.
We have cr( K i ) ≤ cr( P ) − for i ∈ { , } .Proof. The diagram δ ∩ D P of the tangle ( B, t ) is formed by two curves, c and c say, corresponding to the components t and t , respectively, of t . We denote by cr( c i ) the number of double points of c i . Since a diagram of K can be obtainedfrom c by adding a simple arc in d , it follows that we have(3) cr( K ) ≤ cr( c ) . Observe that by construction we have(4) cr( P ) = cr( D P ) = cr( c ) + cr( c ) + card( c ∩ c ) . By the definition of ρ (this definition is given at the beginning of Sec. 6) we have(5) ρ ( I, J ) = min { c ) + card( c ∩ c ) , c ) + card( c ∩ c ) } . N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 13
Since ρ ( I, J ) ≥ P )3 , it follows from (3), (4), and (5) that cr( K ) ≤ cr( c ) ≤
23 cr( P ) − card( c ∩ c )2 . Since D P is a minimal diagram of a prime knot and I = J , it follows that c ∩ c = ∅ . Assuming that c intersects c in a unique point ( q , say) implies that q is a cutpoint of D P . However, no minimal diagram of a knot has a cutpoint.This implies that card( c ∩ c ) ≥ and cr( K ) ≤ cr( P ) − , as required. Thecase of K is analogous. (cid:3) Claim 8.
The -string tangle ( B, t ) represented by the diagram δ ∩ D P is eitherprime or trivial.Proof. Suppose on the contrary that ( B, t ) is neither prime nor trivial. Then ( B, t ) is locally knotted, that is, B contains a ball A such that the pair ( A, A ∩ t ) is a nontrivial -string tangle. Let t i , where i ∈ { , } , be the component of t that meets A . We denote by L the knot that is the closure of the -string tangle ( A, A ∩ t ) . Then L is a factor of P . Since P is prime and L is nontrivial, itfollows that L and P are equivalent. At the same time, L is a factor of K i (asdefined above, K i is the closure of the -string tangle ( B, t i ) ). Since L and P areequivalent, while P is assumed to be -regular, we have cr( K i ) ≥ cr( P ) , whichcontradicts Claim 7. The obtained contradiction proves that ( B, t ) is either primeor trivial. (cid:3) Thus, all requirements from the definition of weak property PT are fulfilled.Consequently, D P has weak property PT. Proposition 3 is proved. (cid:3) Proposition 4. If P is a knot with weak property PT, then there exists a prime γ -knot P ′ over P with cr( P ′ ) ≤ cr( P ) + 17 .Proof. By definition, P has a minimal crossing diagram D P with weak prop-erty PT. This means that there exists a disk d ⊂ S such that– the boundary ∂d intersects D P transversely in four points;– the intersection d ∩ D P consists of two simple non-intersecting arcs;– the tangle diagram δ ∩ D P , where δ := S \ int( d ) , represents either prime ortrivial -string tangle ( B, t ) .Without loss of generality we can identify the pair ( d, d ∩ D P ) with the tanglediagram in Fig. 1(a). We have the following two cases:( α ) two arrows on the arcs in Fig. 1(a) indicate the same orientation on P ,( β ) two arrows on the arcs in Fig. 1(a) induce opposite orientations on P .In case ( α ), let D α be the diagram obtained from D P by local move as inFig. 1 and let P α be the knot represented by D α . Since the figure-eight knot ishyperbolic, an easy argument shows that P α is a γ -knot over P . We check that P α has all of the desired properties. First, the obtained diagram D α of P α has cr( P ) + 16 crossings. This means that cr( P α ) ≤ cr( P ) + 16 . Next, we prove that P α is prime. We observe that, by construction, P α is a sum of the cable tangleof Fig. 1(b) and the tangle ( B, t ) , which is prime or trivial. Each nontrivial cabletangle is prime (see Lemma 3). If ( B, t ) is prime then P α is prime by Theorem 2.If ( B, t ) is trivial then P α is prime by Lemma 4 and assertion (i) of Proposition 1(Lemma 4 implies that P α is either prime or trivial if ( B, t ) is trivial; assertion (i) A point x of a connected topological space X is a cutpoint if the set X \{ x } is not connected. implies that P α is a satellite knot and hence nontrivial). Thus, P α is a prime γ -knot over P with cr( P α ) ≤ cr( P ) + 16 , as required.In case ( β ), let D β be the diagram obtained from D P by local move as in Fig. 4and let P β be the knot represented by D β . Figure 4.
Type I Reidemeister move plus double figure-eight moveThe local move in Fig. 4 is the composition of a type I Reidemeister move andthe move shown in Fig. 1. This implies that P β is a γ -knot over P . Obviously, D β has cr( P ) + 1 + 16 crossings. This means that cr( P β ) ≤ cr( P ) + 17 . Theprimeness of P β follows by the same argument as in case ( α ) because P β is a sumof a nontrivial cable tangle and the tangle ( B, t ) . Thus, P β is a prime γ -knot over P with cr( P β ) ≤ cr( P ) + 17 , as required. (cid:3) Assertion (iii) of Proposition 1 readily follows from Proposition 4 by Proposi-tion 3. 9.
Addendum I: Strong property PT
In addition to weak property PT defined in Sec. 8, we introduce strong prop-erty PT.
Definition. Strong property PT.
Let D be a knot diagram on the -sphere S = R ∪ {∞} . We say that a tangle ( B, t ) is represented by a connected sub-diagram of D if there exists a -disk δ ⊂ S such that the intersection δ ∩ D isconnected and the pair ( δ, δ ∩ D ) , with information of under- and overcrossingsinherited from D , is a diagram of ( B, t ) . We say that D has strong property PT ifevery -string tangle represented by a connected subdiagram of D is either primeor trivial. We say that a knot has strong property PT if all of its minimal diagramshave strong property PT. Proposition 5.
1. Each minimal diagram of each -regular prime knot has strongproperty PT. In particular, each -regular prime knot has strong property PT.2. Each minimal diagram with strong property PT has weak property PT.In particular, each knot with strong property PT has weak property PT.Proof.
1. Assume to the contrary that a non-prime non-trivial -string tangle ( B, t ) is represented by a connected subdiagram δ ∩ D P in a minimal diagram D P of a -regular prime knot P . This implies in particular that ( B, t ) is locallyknotted, that is, B contains a ball B ′ such that ( B ′ , B ′ ∩ t ) is a nontrivial -stringtangle. Let K denote the knot obtained by the closure of ( B ′ , B ′ ∩ t ) . Then K is a factor of P , which is a prime knot, so that we have K = P . (This follows by N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 15 the Unique Factorization Theorem by Schubert [Schu49].) On the other hand, theknot K = P is a factor of the knot K obtained as the closure of ( B, t ) , where t is the component of t that meets B ′ . Observe that we have cr( K ) ≤ cr( P ) − because, since the diagram δ ∩ D P representing ( B, t ) is connected, the projectionof t has at least one crossing with the projection of the second component of t .The inequality cr( K ) ≤ cr( P ) − implies that K = P . Therefore, K is acomposite knot, P is a factor of K , and cr( K ) ≤ cr( P ) − . This contradictsthe assumption that P is a -regular knot.2. Let D be a minimal diagram with strong property PT. If D is a circle withno double points then D has weak property PT (obvious). Assume that D hasdouble points. We take a double point x of D and consider a disk d ⊂ S ina small neighborhood of x such that the intersection d ∩ D consists of two non-intersecting arcs (as on the left side of Fig. 1). Since D is a minimal diagram, x is not a cutpoint of D . This easily implies that the intersection δ ∩ D , where δ := S \ int( d ) , is connected. Since D has strong property PT, it follows that the -string tangle represented by the connected subdiagram δ ∩ D is either prime ortrivial. This means that D has weak property PT. (cid:3) Propositions 5 and 3 give the following dependence for properties of primeknots. -regularity = ⇒ -regularity ⇒ ⇒ strong property PT = ⇒ weak property PTThis implications can be treated in terms of conjectures. We consider thefollowing conjectures. Conjecture 7.
Each prime knot has strong property PT.
Conjecture 8.
Each prime knot has weak property PT.
Conjecture 9.
There exist ε > and N > such that, for all n > N , thepercentage of knots with weak property PT amongst all of the hyperbolic knotsof n or fewer crossings is at least ε . We have the following implications.Conj. 4 = ⇒ Conj. 5 = ⇒ Conj. 6Conj. 2 = ⇒ Conj. 3 = ⇒ Conj. 7 = ⇒ Conj. 8 = ⇒ Conj. 9 = ⇒ = ⇒ = ⇒ The implication Conj. 4 ⇒ Conj. 7 follows from assertion 1 of Proposition 5.The implication Conj. 7 ⇒ Conj. 8 follows from assertion 2 of Proposition 5. Theimplications Conj. 5 ⇒ Conj. 8 and Conj. 6 ⇒ Conj. 9 follow from Proposition 3.The implication Conj. 8 ⇒ Conj. 9 is obvious.Theorem 1 can be strengthened in the following way.
Theorem 3.
Conjecture 1 contradicts (each of ) Conjectures 2–9.
Proof.
Since each of Conjectures 2–8 implies Conjecture 9 (see the system of im-plications before Theorem 3), it suffices to show that Conjecture 1 contradictsConjecture 9. In order to prove this, we repeat verbatim the reduction of Theo-rem 1 to Proposition 1 up to replacing -regularity with weak property PT andassertion (iii) of Proposition 1 with Proposition 4. (cid:3) Addendum II: Non- -regular knots The main theorem of the present paper states that Conjecture 1 concerningpredominance of hyperbolic knots contradicts the conjecture on additivity of thecrossing number (of knots under connected sum) as well as several weaker conjec-tures. In this section, we show that Conjecture 1 also contradicts an assumptionthat the conjecture on additivity has many strong counterexamples.We say that a knot P is non- λ -regular , λ ∈ R , if there exists a knot K suchthat P is a factor of K while cr( K ) < λ · cr( P ) . In this section, we prove thefollowing theorem. Theorem 4.
If there exist ε > and N > such that, for all n > N , thenumber of non- -regular knots of n or fewer crossings is at least ε H n , where H n is the number of hyperbolic knots of n or fewer crossings, then Conjecture 1 doesnot hold.Proof. Suppose that the assumption of the theorem holds true, denote by M the set of all non- -regular knots, and let f be a map with domain M sending K ∈ M to a composite knot f ( K ) with factor K such that(6) cr( f ( K )) <
14 cr( K ) . Then the result of Lackenby [La09] stating that for any knots K , . . . , K n in the -sphere we have(7) cr( K ) + · · · + cr( K n )152 ≤ cr( K ♯ . . . ♯K n ) implies that for each knot L in the codomain f ( M ) we have(8) card( f − ( L )) < / . Indeed, let K be a knot with f ( K ) = L having the smallest crossing numberamong the elements of f − ( L ) . Then (7) implies that(9) card( f − ( L )) cr( K )152 ≤ cr( L ) . Obviously, (6) and (9) imply (8).Since all of the knots in f ( M ) are composite, it follows by (6) and (8) thatfor all n ∈ N we have(10) card { K ∈ M : cr( K ) ≤ n } < card { L ∈ f ( M ) : cr( L ) ≤ n } ≤ C n , where C n is the number of composite knots of n or fewer crossings. At the otherhand, by the assumption of the theorem, for all m > N we have(11) H m ε ≤ card { K ∈ M : cr( K ) ≤ m } . N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 17
Then (10) and (11) imply that for all n > N / we have(12) ε H n < C n . Now, we observe that each knot K in the -sphere obviously has a two-strandcable knot J K with cr( J K ) ≤ K ) + 1 . Since a cable knot over a nontrivialknot is a prime satellite knot (see [Schu53, p. 250, Satz 4], [Gra91, Cor. 2]), whilecable knots over distinct knots are distinct (Lemma 5 below), it follows by (12)that for all n > N / we have ε H n < C n < S n +1 , where S m denotes the number of all prime satellite knots of m or fewer cross-ings. Consequently, since the sequences ( H i ) i ∈ N and ( S i ) i ∈ N are monotonicallyincreasing, for all m > N we have ε H m < S m +4 . As is shown in Section 3 (see deduction of Theorem 1 from Proposition 1), con-ditions of this kind contradict Conjecture 1. (cid:3)
Lemma 5.
Cable knots over distinct knots are distinct.Proof.
By Corollary 2 of [FW78], the group of a cable knot J ( p, q ; K ) determinesthe numbers | p | and | q | and the topological type of K ’s complement. By theGordon–Luecke theorem [GL89], the knot complement determines the knot. (cid:3) Addendum III: Weak property PT and unknotting numbers
This section deals with a relation between weak property PT and the unknot-ting number of knots. The unknotting number of a knot K is denoted by u ( K ) . Definitions.
Let us say that a knot P is weakly U-regular if we have u ( P ) ≤ u ( K ) whenever P is a factor of a knot K . We say that a knot P is strictly U-regular if we have u ( P ) < u ( K ) whenever P is a factor of a knot K = P . We say thata knot P has weak BJ-property if by altering one of the crossings in a minimaldiagram of P we obtain a knot J = P with u ( J ) ≤ u ( P ) . We say that a knot P has strict BJ-property if by altering one of the crossings in a minimal diagramof P we obtain a knot J with u ( J ) < u ( P ) . Remarks.
1. The conjecture that all knots are strictly U-regular is weaker thanthe old conjecture on additivity of the unknotting number of knots under con-nected sum (see, e. g., [Ad94b, p. 61], [Kir97, Problem 1.69]). At the moment, nocounterexample seems to be known to the latter conjecture. Thus, no examples ofnon-U-regular knots are known up to now. The theorem of Scharlemann [Scha85]saying that unknotting number one knots are prime (together with the UniqueFactorization Theorem by Schubert [Schu49]) implies that all knots with unknot-ting number one are strictly U-regular, while all knots with unknotting numbertwo are weakly U-regular.2. The so-called Bernhard–Jablan conjecture (see [Be94], [Ja98], and [JS07]) isequivalent to the conjecture that all knots have strict BJ-property. Kohn’s con-jecture [Koh91, Conjecture 12] (which can be viewed as a particular case of theBernhard–Jablan conjecture) is equivalent to the conjecture that all knots with unknotting number one have strict BJ-property. The set of knots with strict BJ-property contains the set of knots satisfying the Bernhard–Jablan conjecture. Atthe moment, no counterexample seems to be known to the Bernhard–Jablan con-jecture. Available results concerning unknotting number shows that many smallknots and some specific classes of knots satisfy the Bernhard–Jablan conjecture,hence have strict BJ-property. For example, results of [KrM93] and [Mur91] implythat all torus knots have strict BJ-property. Results of McCoy [McC13] implythat alternating knots with unknotting number one have strict BJ-property.
Proposition 6.
1. Each weakly U-regular prime knot with strict BJ-property hasweak property PT.2. Each strictly U-regular prime knot with weak BJ-property has weak prop-erty PT.Proof. If P is a weakly [resp., strictly] U-regular prime knot with strict [resp.,weak] BJ-property, then there exists a minimal diagram D P of P (on the -sphere S = R ∪ {∞} ) with a crossing X such that the change of the crossingyields a diagram of a knot J with u ( J ) = u ( P ) − [resp., a knot J = P with u ( J ) ≤ u ( P ) ]. Let d be a disk in S containing x such that the intersection d ∩ D P is homeomorphic to × while ∂d intersects D P transversally in four points.Let δ denote the disk S \ int( d ) , and let ( B, t ) be the -string tangle representedby the diagram δ ∩ D P .We show that ( B, t ) has no local knots. Suppose on the contrary that ( B, t ) is locally knotted, that is, B contains a ball A such that the pair ( A, A ∩ t ) isa nontrivial -string tangle. We denote by L the knot that is the closure of the -string tangle ( A, A ∩ t ) . Then L is a factor of P . Since P is prime and L isnontrivial, it follows that L and P are equivalent. At the same time, L (= P ) is a factor of J . Then we have u ( P ) ≤ u ( J ) because P is weakly U-regular[resp., u ( P ) < u ( J ) because P is strictly U-regular while J = P ]. However, u ( J ) = u ( P ) − [resp., u ( J ) ≤ u ( P ) ]. The obtained contradiction proves that ( B, t ) has no local knots.Now, we take a subdisk d ′ in d such that the intersection d ′ ∩ D P consists of twosubarcs on two distinct legs of × = d ∩ D P (while ∂d ′ intersects D P transverselyin four points): d ′ Let δ ′ denote the disk S \ int( d ′ ) . Obviously, the diagram δ ′ ∩ D P representsthe same -string tangle ( B, t ) , which has no local knots. Thus, the requirementsfrom the definition of weak property PT are fulfilled. Consequently, P has weakproperty PT. (cid:3) Corollary 2.
If there exist ε > and N > such that, for all n > N , thepercentage of weakly U-regular knots with strict BJ-property amongst all of thehyperbolic knots of n or fewer crossings is at least ε , then Conjecture 1 does nothold. N THE QUESTION OF GENERICITY OF HYPERBOLIC KNOTS 19
Proof.
By Proposition 6, the assumption of the corollary implies Conjecture 9(which concerns the set of knots having weak property PT). By Theorem 3, Con-jecture 9 contradicts Conjecture 1. (cid:3)
Corollary 3.
If there exist ε > and N > such that, for all n > N , thepercentage of strictly U-regular knots with weak BJ-property amongst all of thehyperbolic knots of n or fewer crossings is at least ε , then Conjecture 1 does nothold.Proof. See the proof of Corollary 2. (cid:3)
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