Some knots with surgeries yielding lens spaces
John Berge


Some Knots With Surgeries Yielding Lens Spaces
John Berge


This is a facsimile of the circa 1990 unpublished manuscript with the same title. All the original text, figures and tables are included; although text has been reset in , the original hand-drawn figures have been redrawn digitally, and the parameter k in the original table of lens spaces has been replaced with the originally intended λ. And, of course, this abstract has been added.




Version 1.0





We describe several families of knots in S^3 which have non-trivial Dehn-surgeries
yielding lens spaces. These knots are characterized by Theorem 1 and appear to
include all known knots in S^3 which have surgeries yielding lens spaces.

Theorem 1.
Let k be a simple closed curve on the boundary of a standardly embedded genus two
handlebody H in S^3. If k represents a free generator of both the fundamental group of H and the fundamental group of the complementary handlebody H^' in S^3, then there is an integer surgery on k which yields a lens space.


Proof.
Observe that because k represents a free generator of π_1(H), there exists a set
of cutting disks D_1, D_2 for H with the property that k intersects D_1 exactly once transversely, and k is disjoint from D_2. Similarly, there exists a set of cutting disks D_1^', D_2^' for H^' with the property that k intersects D_1^' exactly once transversely, and k is disjoint from D_2^'. Note that adding the 2-handle D_2^' to H yields a manifold with torus boundary embedded in S^3. Thus this manifold is the exterior of a knot in S^3. Note also that adding the 2-handle D_1^' to the boundary of this manifold yields S^3. Hence D_1^' is the meridian of this knot. Next, note that because k is transverse to D_1^' at a single point, surgery on this knot along k is an integer surgery. Observe also, since k intersects D_1 only once, that adding a 2-handle to H along k yields a solid torus. Thus surgery on this knot along k yields a manifold which has a Heegaard splitting of genus one and which must therefore be a lens space. Finally, note because k is transverse to D_1^' at a single point, k is isotopic to the core of this knot, and so we may take k itself as representing the knot on which the surgery is performed. Q.E.D.

Remark.
We will sometimes informally refer to a curve k satisfying Theorem 1 as a 
double-primitive and to the knot k as a double-primitive knot. We also mention that since the surgery curve k has the same knot type as the knot itself, we will often blur the distinction between the surgery curve and the knot.

Given a knot k in S^3 which has an integer surgery that yields a lens space, it is
in general not easy to determine whether k satisfies Theorem 1. Of course, if
k satisfies Theorem 1, k must be a tunnel-number-one knot, but in general a tunnel-number-one knot will have many inequivalent tunnels and finding all of these tunnels and checking whether any of them lead to a decomposition of the exterior of k such that the surgery curve on k satisfies Theorem 1 is not an easy matter. Hence we sought to find more explicit descriptions of the knots in S^3 satisfying Theorem 1 and to find explicit formulas giving the lens spaces that result from surgeries on these knots. These descriptions and formulas should be useful in determining whether a knot with surgery yielding a lens space satisfies Theorem 1 and if it does satisfy Theorem 1 they should be useful in precisely identifying the knot.

We also show that the knots satisfying Theorem 1 correspond to knots in lens spaces which are either 0 or 1-bridge knots in lens spaces and which have integer surgeries yielding S^3. This alternative characterization may also prove useful in identifying knots of this type, and we describe it below.

We must also include a warning that there is a chance that the list of explicit descriptions
of the knots satisfying Theorem 1, which we give, is not complete. We believe that there is fairly good evidence that it is complete but we do not have a proof of this fact. 

Thanks to Professor Jim Cannon and my major professor D.R.McMillan Jr. for their patience and support.

This leaves open the possibility that there are knots that satisfy Theorem 1 which are not on any of the lists we give.

At the end of the paper, we have included a table which gives those lens spaces L(p,q) with p ≤ 500 that are obtainable via surgeries on the knots in our list. However, to keep the size of the table small, we have not included those lens spaces which are obtainable only via surgeries on torus knots or cables about torus knots.

Introductory Definitions and Lemmas.


A Heegaard splitting (M,F), of a closed, connected, orientable 3-manifold M is a closed, connected, orientable surface F embedded in M which separates M into two components whose closures are handlebodies V and W. If (M,F) is a Heegaard splitting, then a choice of complete sets, say v and w, of meridional disks for V and W yields a Heegaard diagram (F,∂ v,∂ w), of (M,F). We will also use the notation (F,v,w) for a Heegaard splitting when M, V and W are understood and our interest is in F and sets of cutting disks 
v for V and w for W. We always assume, unless indicated otherwise, that the number of intersections of ∂ v with ∂ w is minimal up to isotopy in F.

It is known that genus one Heegaard splittings of lens spaces are unique up to isotopy [].
Thus if L(p,q) is a lens space, then H_1(L(p,q)) has a pair of distinguished generators which are represented by the cores of the two solid tori of a genus one Heegaard splitting of L(p,q). Here, and throughout the paper, we use integral homology. Let k be a knot in L(p,q) and g one
of the distinguished generators of H_1(L(p,q)). Note that if [k]=λ[g] and (λ,p) = 1, then λ is determined up to ±λ^± 1  p. Thus, ±λ^± 1 p is an invariant of k. We will have use for this invariant later.

Let M be a lens space. A knot k in M is a 0-bridge knot if k is isotopic to a knot k^' lying in a torus which is a Heegard surface of a genus one Heegaard splitting of M. A knot k in M is a 1-bridge knot in M if k is not a 0-bridge knot in M and k is isotopic to a knot k^' which intersects each of the solid tori of a genus one Heegaard splitting of M in a single unknotted arc.

If k is a knot in a 3-manifold M, the exterior of k is the complement of an open tubular neighborhood of k in M. Note that if k is a knot in manifold M and integer surgery on k yields a manifold N, then there is an associated knot k' in N whose exterior is homeomorphic to the exterior of k and which has an integer surgery yielding M.

Lemma 1.
Let k be a knot in S^3 which satisfies Theorem 1, suppose that integer surgery on k yields a lens space M, and let k^' be the associated knot in M, then k^' is 0-bridge or 1-bridge in M.

Proof.
We use the notation used in the proof of Theorem 1. Observe that when a 2-handle is added to H along the surgery curve k, H becomes a solid torus V and ∂ V is a genus one Heegaard splitting surface for M. Let V^' be the other solid torus of this genus one Heegaard splitting. Let A be an annular neighborhood of the surgery curve k in ∂ H. Note ∂ D_1^' represents the 
associated knot k^' in M and ∂ D_1^'∩ A is a subarc of ∂ D_1^' which is an essential arc α in A. Let β be the subarc arc of ∂ D_1^' that does not lie in int(A). Then β is an unknotted arc which lies in ∂ V, and so it is certainly unknotted in V^'. 

Next consider the subarc α of ∂ D_1^'. Let γ denote the arc ∂ D_1 ∩ A. Then γ is an essential arc in A. And we may assume that A has been chosen so that α and γ are disjoint.
Then α is parallel into the boundary of V via an isotopy which pushes the interior of α across a disk in H isotopic to D_1, while keeping the endpoints of α fixed. In other words, α is unknotted in V. Thus the knot k^' is the union of two arcs, α and β, intersecting only in their endpoints, with α unknotted in V and β unknotted in V^'. So k^' is 0-bridge or 1-bridge in M. Q.E.D.

Lemma 2.
Let (F,∂ v,∂ w) be a genus two Heegaard diagram of S^3 and suppose that k is a simple closed curve in F such that k intersects ∂ v_1 transversely exactly once, k intersects ∂ w_1 transversely exactly once and k is disjoint from ∂ v_2 and ∂ w_2. Suppose each curve ∂ v_1,∂ v_2,∂ w_1, and ∂ w_2 is oriented, then all intersections
of ∂ v_2 with ∂ w_2 have the same sign. If, in addition, there is no edge of the diagram with its endpoints on ∂ v_1 or ∂ w_1, then the curves of the diagram can be oriented so that all intersections of ∂ v_n with ∂ w_m have the same sign for n,m ∈{1,2 }. 

Proof
First note that because k intersects the cutting disks of each handlebody only once, there are no edges of the diagram (F,∂ v,∂ w) which have both endpoints on the same side of ∂ v_2 or ∂ w_2. Next, note that if there are edges of the diagram with both endpoints on the same side of ∂ v_1 or ∂ w_1,
then we can replace ∂ v_1 or ∂ w_1 with a new cutting disk and obtain a new diagram with fewer intersections and still maintain the property that k intersects the cuttings disks in each handlebody only once. By [] if this diagram is nontrivial, then it has a wave ω. This now forces the claimed result as can easily be seen by examining the possible diagrams that can occur. Q.E.D.
  
Lemma 2 allows us to show that a 1-bridge knot in a lens space M, which has a Dehn-surgery yielding S^3, is embedded in M in a particularly simple and pleasant manner.

Theorem 2.
Let (F, v, w) be a genus one Heegaard splitting of a lens space M and let k be a 1-bridge knot in M such that Dehn-surgery on k yields S^3. Assume that ∂ v and ∂ w intersect minimally up to isotopy in F. Then k is isotopic to a knot k^' in M such that k^' is the union of two arcs α and β, with α properly embedded in v and β properly embedded in w.

Proof.
Since k is 1-bridge in M, we may assume that k is the union of two arcs α and β with α unknotted in V and β unknotted in W. It follows that we may also assume that β lies in F and that α is properly embedded in V with α disjoint from v. Note that if β is not disjoint from w, then there is an isotopy of w which makes w disjoint from β.

Let N be a small regular neighborhood of α in V. Then V minus the interior of N is a handlebody H of genus two. Let A be the annulus in which N meets H, and let m be an essential simple closed curve in A. Note that m represents a primitive element in π_1(H) and there is a cutting disk v^' for H, disjoint from v, such that ∂ v^' intersects m transversely exactly once.

Now consider doing surgery on k to get S^3. Since k is not 0-bridge, it follows that the exterior of k is not Seifert-fibered. Since M is irreducible, and surgery on k yields S^3, the exterior of k must be irreducible. Then the Cyclic-Surgery-Theorem of [C,G,L,S] shows that the surgery which yields S^3 must be an integer surgery.

Next, note that the exterior of k is homeomorphic to H ∪ w. Let c be a simple closed curve on ∂ H to which a 2-handle is added to perform the surgery. Then c intersects m transversely exactly once. Thus we have a genus two Heegaard splitting of S^3 with m intersecting the cutting disks of each handlebody once. Lemma 2 then implies that if ∂ v and ∂ w are oriented, the intersections of ∂ v and ∂ w all have the same sign. This in turn implies that k
is isotopic to a knot of the claimed form in M. Q.E.D.

Remark.
Observe that Theorem 2 shows that a 1-bridge knot, in a lens space M, which has a surgery yielding S^3, is isotopic to a knot which is simultaneously braided with respect to both of the solid tori of a genus one Heegaard splitting of M.
  
The next theorem, which follows from Theorem 2, shows that 1-bridge knots in a lens space, which have surgeries yielding S^3, are distinguished by the homology classes that they represent.

Theorem 3.
Suppose that k and l are 1-bridge knots in a lens space M such that surgery on both k and l yields S^3 and k is homologous to ±l in M. Then k is isotopic to ±l in M.

Proof
The proof is easy and follows from the fact that k and l are isotopic to knots with the form shown to exist in Theorem 2. Q.E.D.

The following formula allows one to determine the lens space that results from performing surgery along a surgery curve that represents a double primitive knot in Theorem 1, provided one knows the homology class that the surgery curve represents in the boundary of the handlebody H.

Choose a standard basis for H_1(∂ H) as in Fig. 1 with a,b bounding disks in H 
and A,B bounding disks in H^'.

Insert Fig. 1 here.

Lemma 3.
Suppose k is a simple closed curve in ∂ H satisfying Theorem 1. If 

    [k] = A[A]+B[B] +a[a]+b[b]

in H_1(∂ H), then surgery along k, on the knot represented by k, yields a lens space
L(p,q), with

    p    = |Aa+Bb|, 
    
    a^2q    ≡± B^± 2 p,  and 
    
    b^2q    ≡± A^± 2 p.


Proof.
Let h be an orientation preserving homeomorphism of H^' such that h(a)=k.
Then h induces a sympletic transformation ϕ on H_1(∂ H^'). Since intersection numbers are preserved, we must have:

    ϕ(A)    = (-s,-r,-0,-0), 
    ϕ(B)    = (-b,-a,-0,-0), 
    ϕ(a)    = (aA,bB,-a,-b), 
    ϕ(b)    = ( -i,-j,-r,-s)

where i,j,r, and s are integers satisfying as-br=1 and Ar+Bs=ai+bj. Now we want to change the basis of H_1(H) so that the surgery curve k represents (1,0) and project everything onto the other generator (0,1) of H_1(H). This can be done by forming the dot-product of ϕ(A), ϕ(B),
ϕ(a), and ϕ(b) with (-B,A,0,0) and projecting to get:

    ϕ^'(A)    = -Ar-Bs, 
    ϕ^'(B)    = -Aa+Bb, 
    ϕ^'(a)    = -0, 
    ϕ^'(b)    = -Aj-Bi

where ϕ^' denotes ϕ composed with projection onto the second coordinate of H_1(H).

Recalling that performing surgery along k turns H into a solid torus V, we have a genus one Heegaard splitting of a lens space L(p,q) with a core of V represented by the generator (0,1) of H_1(H). In addition, ϕ(B) represents a meridian of the other solid torus W of the genus one Heegaard splitting of L(p,q) and ϕ(b) represents a longitude of W. It follows that p= |ϕ^'(B)| = |Aa+Bb|.

Note that the multiple of (0,1) which the core of W represents in H_1(L(p,q)) determines q mod p. So we want to evaluate ϕ^'(b) mod p. We will do this by observing that there is a relationship between q and the homology class which the knot k^' dual to k represents in L(p,q). Since ϕ(A) is homologous to k^' in L(p,q), we may take ϕ^'(A) as a representative of the homology class k^' represents in H_1(L(p,q)).

Let λ = ϕ^'(A) =-(Ar+Bs), and assume, for the moment, that neither a nor B is congruent to zero mod p. Then aλ =-(Aar+Bas). Or, since, from the expression for p above, Aa≡ -Bb mod p, aλ≡ -B(as-br) mod p. But, since as-br=1, aλ =-B mod p. Then if (a,p) = 1, λ≡ -B/a mod p.

Next consider the equation Ar+Bs=ai+bj. Replacing Ar+Bs with -λ, then multiplying both sides by B and using the fact that Bb ≡ -Aa mod p, yields Bλ≡ a(Aj-Bi) mod p. Then using λ≡ -B/a mod p and replacing Aj-Bi with ϕ^'(b), we have ϕ^'(b) ≡ -λ^2 mod p or equivalently, a^2 ϕ^'(b) ≡ B^2 mod p. Finally, the classification of lens spaces implies ϕ^'(b) ≡± q^±1 mod p . So q ≡±λ^±2 mod p. Or a^2 q ≡± B^±2 mod p. This is the first of the two claimed formulas for q. 

If a or B is congruent to zero mod p, then b is not congruent to zero mod p and A is not congruent to zero mod p. So this derivation can be repeated, with appropriate changes, to obtain alternative expressions λ≡ A/b mod p for λ and b^2 q ≡± A^±2 mod p for q. Q.E.D.

Descriptions of the knots. 


Those knots which lie in a solid torus and have non-trivial surgeries yielding a solid torus are a special case of the class of knots satisfying Theorem 1. The first six families of knots that we describe arise in this way. Next, we describe a family of knots each of which lives on a Seifert surface of a trefoil knot in S^3, and another family of knots each of which lives on a Seifert surface of a figure-eight knot in S^3. Then there are a final four families of knots satisfying Theorem 1, which we call “sporadic”, and we conclude by describing them.

The knots arising from knots in solid tori with surgeries yielding solid tori.


I) Torus Knots.

Let k be the simple closed curve in ∂ H as illustrated by the R-R diagram in Fig. 2.

A=1, (B,b)=1, and B ≥ 2.

Insert Fig. 2 here.

ii) Cables about Torus Knots.

Let k be the simple closed curve in ∂ H as illustrated by the R-R diagram in Fig. 3. 

A=1, (B,b)=2, and B≥ 4.

Insert Fig. 3 here.

III) 	Let k be the simple closed curve in ∂ H obtained by twisting the simple closed curve c J times to the left about the curve T, as illustrated by the R-R diagram in Fig. 4.

Insert Fig. 4 here.


One checks that, if J ≥ 0, n≥ 0, ε =± 1 and K is arbitrary, then k satisfies Theorem 1. Let A=J+1, B=(J+1)(2n+ε), a=± 1, and b=-a(2ε(J+1)+BK). Then [k] has coordinates:

    [k] = (A,B,a,b)
 in H_1(∂ H). The constraints on J, n, and ε become the following constraints on A, B, a, and b: A>1, there exists ε =± 1 such that (B+ε)/A is an odd integer, a=± 1, and b≡ -2ε Aa mod B.

IV)	Let k be the simple closed curve in ∂ H obtained by twisting the simple closed curve c J times to the right about the curve T, as illustrated by the R-R diagram in Fig. 5.

Insert Fig. 5 here.

One checks that if J ≥ 0, n≥ 0, ε=± 1, and K is arbitrary, then k satisfies Theorem 1. Let A=2J+1, B=2n(J+1)+Jε), a=± 1, and b=-a(ε(2J+1)+BK). Then [k] has coordinates:
[k] = (A,B,a,b)  in H_1(∂ H). The constraints on J, n, and ε become the following constraints on A, B, a, and b:
A>3, there exists ε=± 1 such that (2B+ε)/A is integral, a=± 1, and b≡ - ε Aa mod B.

V)	Let k be the simple closed curve in ∂ H obtained by twisting the simple closed curve c J-1 times to the right about the curve T, as illustrated by the R-R diagram in Fig. 6.

Insert Fig. 6 here.

One checks that, if J≥ 0, n ≥ 0, ε=± 1, and K is arbitrary, then k satisfies Theorem 1. Let A=2J+1, B=n(2J+1)+Jε, a=± 1, and b=-a(ε(2J+1)+BK), then [k] has coordinates:
[k] = (A,B,a,b) in H_1(∂ H). The constraints on J, n, and ε become the following constraints of A, B, a, and b:
A>1 and odd, there exists ε=± 1 such that (B-ε)/A is integral, a=± 1, and b ≡ -ε Aa mod B. 

VI)	Let k be the curve obtained by twisting the curve c J times to the left about the curve T as illustrated in Figure 7 below. 

Insert Fig. 7 here.

Then [k] = (A,B,a,b), where A>2, A is even, B=2A+1, and b≡ a(A-1) mod B with J ≥ 1.

A family of knots obtained from a Seifert surface of a trefoil knot.

Let g_1 and g_2 be simple closed curves embedded in ∂ H as in Figure 8.

Insert Fig. 8 here.


We mention that if N is a regular neighborhood of g_1∪g_2 in ∂ H, then ∂ N
is a trefoil knotted curve c in S^3 and N is a Seifert surface for c. In the chosen homology basis for ∂ H, we have [g_1]=(1,0,-1,0) while [g_2]=(0,1,-1-1). Let k be a nonseparating
simple closed curve in N with 
[k]=m[g_1]+n[g_2]. 
Where, since k is a simple closed curve, (m,n)=1. Note that the handlebody H is homeomorphic to N× I, and that the handlebody H^' is also homeomorphic to N× I. Thus any nonseparating simple closed curve in N represents a free generator of both π_1(H) and π_1(H^'). It then follows from Theorem 1 that there is an integer surgery on k yielding a lens space. Since [k] has coordinates: [k] = (m,n,-(m+n),-n)
in H_1(∂ H), it follows that the lens spaces obtained by integer surgeries on the knots in this family are of the form:
 L(p,q) = L(m^2+mn+n^2,(m/n)^2).

Remark.
Observe that if m or n equals 0, then k is unknotted, while if |m| or |n| equals 1, then k is a torus knot. Hence, we may assume, without loss of generality, that |m| and |n| are greater than 1.


Theorem 4.
A lens space L(p,q) is obtainable via an integer surgery on a knot embedded as a nonseparating simple closed curve in a genus one Seifert surface of a trefoil knot if and only if p is odd, not divisible by 9, every prime factor of p, not equal to 3, is congruent to 1 6, and q≡±((±-1±√(-3))/2)^± 2 p.

Proof.
Consider the quadratic form Ax^2+Bxy+Cy^2. Recall from the theory of quadratic forms that the discriminant of such a form is B^2-4AC and that two quadratic forms in x and y are equivalent if replacing x with ax+by and y with cx+dy, where ad-bc=1, in one form carries that form into the other.

Note that x^2+xy+y^2 is a quadratic form with discriminant -3. If setting x=a and y=b yields p, with (a,b)=1, then we see that it is necessary that -3 be a quadratic residue 4p. By applying the law of quadratic reciprocity, we see that the stated conditions on p are exactly the conditions necessary for -3 to be a quadratic residue 4p. On the other hand, it can be shown that these necessary conditions on p are also sufficient conditions for p to be expressible by some quadratic form of discriminant -3.
However, it is known that any two quadratic forms of discriminant -3 are equivalent [].
It follows from this fact that the conditions on p are also sufficient to guarantee that p can be expressed as p=a^2+ab+b^2 with (a,b)=1.

If p=a^2+ab+b^2 with (a,b)=1, let λ≡ a/b mod p. Then λ satisfies the quadratic equation λ^1+λ+1≡ 0 mod p. Thus λ≡ (-1±√(-3))/2 mod p. Note that, for a given value of
√(-3) mod p, the two possible values of λ are reciprocals p. Finally, by Lemma 3, q ≡±(λ)^± 2 mod p. Q.E.D.

A family of knots obtained from a Seifert surface of the figure-eight  knot.

Let g_1 and g_2 be simple closed curves embedded in ∂ H as illustrated by Fig. 9.

Insert Fig. 9 here.

Then, if N is a regular neighborhood of g_1∪g_1 in ∂ H, ∂ N is a
figure-eight knotted curve c in S^3 and N is a Seifert surface for c. In the chosen homology basis for ∂ H, we have [g_1]=(1,0,-1,0), while [g_1]=(0,1,-1,1). Let k be a nonseparating
simple closed curve in N with
[k]= m[g_1]+n[g_2].
Where, since k is a simple closed curve, (m,n)=1. Note that the handlebody H is homeomorphic to N × I, and the handlebody H^' is also homeomorphic to N × I. Thus any nonseparating simple closed
curve in N represents a free generator of both π_1(H) and π_1(H^'). It then follows from Theorem 1 that there is an integer surgery on k yielding a lens space. Since [k] has coordinates:
[k]= (m,n,-(m+n),n) in H_1(∂ H), it follows that the lens spaces obtained by integer surgeries on the knots in this family are of the form: L(p,q)=L(|n^2-mn-m^2|,(m/n)^2).

Remark.
Observe that, as was the case for the family of knots based on a Seifert surface of a trefoil knot, if m or n equals 0, then k is unknotted, while if |m| or |n| equals 1, then k is a torus knot. Hence we may assume, without loss of generality, that |m| and |n| are greater than 1.


The following theorem characterizes those lens spaces which can be obtained by surgeries on the knots in this family.

Theorem 5.
A lens space L(p,q) is obtainable via an integer surgery on a knot embedded as a nonseparating simple closed curve in a genus one Seifert surface of the figure-eight knot if and only if p is odd, not divisible by 25, every prime factor of p, not equal to 5, is congruent to ± 1 5, and q≡±((-1±√(5))/2)^± 2 p.


Proof.
The expression b^2-ab-a^2 corresponds to a quadratic form of discriminant 5. Thus form can be analyzed in a fashion similar to that used for the form a^2+ab+b^2. We see that the stated conditions on p are necessary in order that p be representable by some quadratic form of discriminant 5. However, it is not hard to check that any two quadratic forms of discriminant 5 are equivalent. It follows from this fact that the conditions on p are also sufficient to guarantee that p can be represented as p=b^2-ab-a^2 with (a,b)=1.

If p=b^2-ab-a^2 with (a,b)=1, let λ = a/b  p. Then λ satisfies the quadratic equation λ^2+λ-1≡ 0 p. Thus λ≡ (-1±√(5))/2 p. Note that, for a given value of √(5) p, the two possible values of λ are negative reciprocals mod p. Finally, by Lemma 3, q=±(λ)^± 2 p. Q.E.D.

Remark.
Note that the construction of the proceeding two families works essentially because each knot in each family is a nonseparating simple closed curve embedded in a genus one Seifert surface of a fibered knot in S^3. If there were other genus one fibered knots besides the left- and right-handed trefoil and the figure-eight knot in S^3, or if the trefoil or the figure-eight knot had other inequivalent genus one Seifert surfaces, then we would expect to have other families of knots, with surgeries yielding lens spaces, obtained from curves embedded in these new Seifert surfaces. However, it is known, that the only fibered knots of genus one in S^3 are the trefoils and the figure-eight, and furthermore, that genus one Seifert surfaces of these knots are unique, up to isotopy []. So the families of knots obtained from Seifert surfaces of the trefoils and the figure-eight knot are the only such families that exist.

The “Sporadic” Knots


Let k be the simple closed curve in ∂ H obtained by twisting the simple closed curve c J times to the right about the curve t, as illustrated by the R-R diagram in Fig. 10.

Insert Fig. 10 here.


Note that in order for k to be embedded in ∂ H, the parameters p, p', m, m', q, q', and n, n' must be chosen so that pm'-mp' = qn'-nq' = 1.
Then observe that k_∗ represents A^pB^n(A^mB^qA^mB^n)^J in π_1(H), and k_∗^ ' represents  a^p^'b^n^'(a^m^' b^q^'a^m^'b^n^')^J in π_1(H^'). Now suppose the ordered set of parameters (p,p^',m,m^', q,q^',n,n^') is replaced by one of the following four sets:

    a)     ( 1,-1,-2,-3,-1,-1,-1,-0 ) 
    b)     ( 2,-1,-3,-2,-1,-1,-1,-0 ) 
    c)     ( 4,-3,-3,-2,-1,-0,-1,-1 ) 
    d)     ( 3,-5,-2,-3,-1,-0,-1,-1 ).


Then, in each of these four cases, k_∗ and k_∗^' represent free generators of π_1(H) and
π_1(H^'), respectively, provided J≥ 0. It then follows from Theorem 1 that, in each case, there is an integer surgery on k yielding a lens space. 

The coordinates of [k] in H_1(∂ H) corresponding to the preceding four possible sets of parameter values are:

    a)      ( 4J+1,12J+1,-6J+1,-J ) 
    b)      ( 6J+2,12J+1,-4J+1,-J ) 
    c)      ( 6J+4,12J+1,-4J-3,-J+1 ) 
    d)      ( 4J+3,12J+1,-6J-5,-J+1 ).


And so, by Lemma 3, the lens spaces obtained by integer surgeries on the knots in these four families are of the form:

    a)      L( 22J^2+19J+11, (11J+2)^2 ) 
    b)      L( 22J^2+13J+12, (11J+3)^2 ) 
    c)      L( 22J^2+31J+11, (11J+8)^2 ) 
    d)      L( 22J^2+35J+14, (11J+9)^2 ).


This completes the list of knots satisfying Theorem 1 known to us. We have not been able to prove that this list is complete. However, there is some computational evidence that it is complete. In particular, John Rosenberg kindly wrote a computer program for us which examined one-bridge knots, in lens spaces with fundamental groups of order less than 1000, for integer surgeries yielding S^3, without finding any knots not on our list.

A presumably more interesting and more difficult problem than characterizing double-primitive knots on genus two Heegaard surfaces in S^3 is that of characterizing all knots in S^3 with Dehn-surgeries yielding lens spaces. Then there is the following natural question.

Question.
If k is a knot in a lens space M such that Dehn-surgery on k yields S^3, must k be a  0 or 1-bridge knot in M?


The following table includes all lens spaces L(p,q) with p≤ 500 which are obtainable via surgeries on knots in our list,
except, to keep the size of the table managable,  those lens spaces which are obtainable only by surgeries on torus knots and cables about
torus knots are not shown. Included with each lens space L(p,q), in column λ, is a list of the homology classes
in L(p,q) that are represented by 1-bridge knots which have Dehn-surgeries that yield S^3. For example, the table shows that L(37,10) contains two such 1-bridge knots.

In order to check whether a lens space L(p,q) with p>500 is obtainable from a knot on these lists, some computation is necessary. Checking whether L(p,q) is obtainable from a torus knot, a cable about a torus knot, a knot
based on a Seifert surface for a trefoil or figure-eight knot or a sporadic knot, is straightforward. To determine whether L(p,q) is obtainable from one of the remaining knots, one can do the following: The relationship p=Aa+Bb holds with
0<2A≤ B and |a|=1. Thus the value of B determines the value of A, a, and b. It then suffices to check those values of B such that q≡±(B)^± 2 p, for 1< B ≤max{8,2p/5}, and to check whether the set of values {A,B,a,b} obtained satisfies the constraints for any of the knots in our list.

Insert Table of Lens Spaces here.


References.








p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ 
 18   5   5         25   142   53   35   189   41   22   226   49   7   269   61   21      135   135 
 
19   7   7   94   35   23   143   25   5      62   62      69   35      70   71   302   83   41 
 
27   8   8   95   39   18   144   47   47      67   32      85   57   270   59   31      109   13 
 
30   11   7      41   42   145   51   52   190   71   47   227   49   7       89   89      113   75 
 
31   11   12   97   35   35      59   28   191   56   18      84   10       101   67   305   121   62 
  
    12   7   98   27   13   146   55   37      87   88   229   64   8       109   53   306   101   101 
 
32   7   5      37   25   147   41   20   192   71   11      80   81   271   28   28      115   77 
 
34   13   9   99   29   13      62   8   193   25   5      94   18       75   14   307   25   5 
 
36   11   11         16      67   67      81   9         94   273   43   20      54   19 
 
37   10   8      32   32   148   41   19      84   84   230   49   13       100   10   308   87   45 
  
      10   100   39   19   149   39   40   194   73   49      91   47          100   309   46   46 
 
39   16   16   101   21   22   150   59   29   195   79   38   231   67   32   274   81   9   311   57   58 
 
43   12   5      30   8   151   26   27   196   55   27   232   25   5       103   69      115   14 
 
45   14   14   103   46   46      32   32      75   9   233   89   12   275   49   7   312   49   7 
  
    19   8   105   41   22      45   14         11   234   35   19       64   8   313   49   7 
 
46   17   11   107   25   5   153   50   50   198   37   17      77   77       109   54      71   23 
 
47   13   9      41   24      55   26      65   65   235   51   24   276   49   7      98   98 
 
49   18   18   108   23   13   155   46   8   199   37   19      54   17       73   35      119   11 
 
50   19   9      35   35      61   32      55   12   236   55   29   277   59   13   315   41   22 
  
       13   109   45   8   157   25   5      60   61   237   55   55       81   9      101   52 
 
54   17   17         45      36   11      92   92      64   8          14      104   104 
 
55   21   12   110   41   27      46   16   200   79   39         10       116   116   317   85   42 
 
57   16   5   112   31   9   158   37   11   201   37   37   238   89   59   278   85   11      121   11 
 
59   24   25         17      59   39   205   36   13   239   70   13   279   83   11         14 
 
61   13   13   114   43   29   161   45   24      46   47      71   36       89   46   318   25   5 
  
    16   17   115   34   9      61   30      61   11   240   71   13       92   92      119   79 
 
62   23   15         14   162   35   17      81   42   241   45   14   280   81   39   319   59   28 
 
63   17   10   116   45   7      53   53   206   63   11      50   51       111   57      138   139 
  
    20   20         25      61   41      77   51   242   43   21   281   36   37      149   150 
 
64   23   19   117   38   38   163   44   10   207   25   5      91   61   282   25   5   320   129   63 
 
66   25   17      43   7      58   58      65   34   243   25   5   283   44   44   322   121   81 
 
67   18   7         20      62   15      68   68      53   26       64   8   323   89   15 
  
    29   29   118   25   5   165   29   16   209   79   80      77   40       78   19   324   49   7 
 
68   19   5   119   50   13      49   7      80   9      80   80       104   20      71   35 
  
       7   120   49   23         8         17   245   69   34   286   51   27      107   107 
 
70   29   13   121   35   36   166   49   7   210   59   31      99   48       107   71   325   49   7 
 
71   21   11      46   14   167   46   11      79   53   247   68   68   287   45   23      129   64 
  
    26   20   125   49   24   168   25   5   211   31   32      87   87      62   15   326   97   11 
 
72   23   23   126   41   41   169   22   22      45   16   250   99   49   288   95   95   327   100   10 
 
73   27   10      47   31   170   69   33      64   8   251   116   117       119   11         11 
 
75   29   11   127   19   19   171   50   11   212   57   27   252   83   83          13      154   154 
  
       14      27   10      53   28      81   9   253   74   20   290   81   9   329   95   46 
 
78   29   19      29   15      56   56   213   59   19   254   95   63       109   73   330   131   67 
 
79   23   23   128   47   7   172   39   21   214   49   7   255   101   52   291   61   61   331   31   31 
  
    28   29         9      63   31   215   49   7   257   25   5   292   111   11      75   16 
  
    29   7   129   49   7   173   64   8   216   49   23      59   14   293   25   5      76   14 
 
80   31   7         49         10      71   71   258   97   65       64   8      115   116 
  
       17   130   49   33   174   65   43   217   25   25   259   73   38       81   9   332   25   5 
 
81   26   26      51   27   175   69   34      67   67      100   100   294   83   41   333   110   110 
  
    31   14   131   50   9   177   49   7   218   25   5      121   121   295   64   22      115   56 
 
82   23   5   132   25   5         17   219   50   13   261   86   86       91   92   334   125   83 
  
    31   21   133   30   30   178   49   7      64   8      91   44       119   58   335   71   34 
 
83   19   8      36   13      67   45         64      100   19   297   67   32   337   63   20 
 
84   25   11      39   18   179   73   74   220   89   43   263   49   7       98   98      91   10 
 
85   26   11   135   31   14   180   59   59   222   83   55      71   10       103   50      94   22 
 
89   34   12      41   22      71   37   223   39   39   264   49   7   300   71   37      128   128 
 
90   29   29      44   44   181   48   48      66   17   265   49   26       119   59   338   51   25 
 
91   16   16   137   37   10      70   16   225   74   74   267   79   13   301   45   16      127   85 
  
    27   8   139   42   42   182   25   5      79   38   268   25   5       64   8   339   64   8 
 
93   25   5      62   63      53   25      89   44      99   13       79   79   340   89   43 
 








p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ   p   q    λ 
 341   74   24   362   49   7      150   151   411   49   7      163   109      118   118   475   64   8 
   
   79   80   363   65   32   391   53   27      64   8   435   91   44   455   81   9      189   94 
  
   100   21      98   10   392   111   55      76   26      94   23      129   66   476   137   67 
  
342   61   29   364   87   45   393   25   5   413   121   11   436   81   9      181   92   477   158   158 
   
   113   113   365   64   8   395   159   78         16      129   13   457   25   5      163   80 
  
343   25   5      69   36      186   187   414   137   137   437   100   10      133   133   478   179   119 
   
   54   17   366   137   91   396   131   131      155   103      169   16      169   13   479   227   228 
   
   97   48   367   83   83   397   34   34   415   54   19         13      72   23   480   191   97 
   
   131   12   368   25   5   398   149   99      116   24      136   14      133   18   481   100   100 
  
345   139   68   369   80   17   399   85   13   416   95   17   438   121   11   459   49   7      211   211 
  
347   61   16      122   122      121   11   417   181   181   439   68   69      55   26   482   25   5 
   
   64   8      127   62         121   418   25   5      78   19      103   50      181   121 
   
   108   14   370   139   93      163   163      157   105      81   9      149   76   484   87   43 
  
349   106   22      149   73   400   119   11   419   95   18      171   171      152   152   485   64   8 
   
   122   122   371   81   9      159   79   420   169   83   440   81   39   460   49   7   486   85   41 
   
   142   143   373   49   7   401   110   111   421   63   22   441   125   62   461   100   19      107   53 
 
350   131   87      88   88      111   17      64   8      146   146   462   173   115      161   161 
   
   139   69      100   10   402   151   101      74   16      151   74   463   63   20   487   91   17 
  
351   77   40   374   49   7   403   61   32      109   110   442   69   33      100   10      105   52 
   
   113   58      81   9      64   8   422   49   7   443   25   5   465   86   28      144   12 
   
   116   116      135   13      87   87   423   49   7      79   27      89   46      232   232 
  
352   97   15   375   149   74      191   191      137   70      129   25   466   129   17   488   73   19 
  
354   133   89   377   144   12   404   105   51      140   140   445   179   88      175   117   489   58   58 
  
355   61   62   378   59   31   405   89   44   425   169   84      186   187   467   64   8   490   139   69 
   
   66   17      85   41      134   134   427   74   74   446   95   13      166   16   491   72   73 
   
   81   9      109   53      139   68      100   10      167   111      193   14   492   119   61 
   
   99   16      125   125      161   82      123   60   447   121   11   468   25   5   493   25   5 
   
   141   72   379   51   51   406   93   47      135   135   448   121   11      121   59      64   8 
  
357   25   5      105   22      115   59   428    103   53   449   164   165      155   155   494   185   123 
   
   64   8   380   71   17   407   25   5   429   64   8   450   59   29   469   37   37   495   161   82 
   
   101   52      151   77      73   38   430   161   107      149   149      71   34      164   164 
 
358   81   9   382   25   5   408   47   25      171   87      169   113      163   163      199   98 
  
359   82   21      143   95      169   13   431   89   90      179   89   470   189   93   498   187   125 
   
   104   105   386   145   97   409   53   53   432   25   5   451   46   47   471   49   7      193   17 
  
360   49   23   387   125   64      120   23      95   49      103   17      169   13   499   89   29 
   
   119   119      128   128      121   11      143   143      125   24         169      139   139 
  
361   41   42   388   163   13         13   433   98   26      156   157   472   49   7      223   224 
   
   49   7         15      128   129      198   198   452   81   9   473   100   10         
   
   68   68   389   91   46   410   49   7   434   121   11   453   104   14      181   14         
 
























email address: jberge@charter.net
