Journal of Knot Theory and Its Ramifications, Vol. 6, No. 6 (1997) 785–798 c
World Scientific Publishing Company
KNOT ENUMERATION THROUGH FLYPES AND TWISTED SPLICES JORGE ALBERTO CALVO Department of Mathematics University of California Santa Barbara, CA 93106
Abstract. I propose a method to enumerate prime alternating knots. The algorithm is both combinatorial and geometrical in nature; it rests on a particular splicing of knot projections (on the sphere S2 ) which is guaranteed to produce every prime knot projection when repeatedly applied (starting on the trivial unknot projection). By exploring the behavior of the flype structure of alternating knots (as in Tait’s Flyping Theorem), I prove not only completeness of the enumerator, but also some useful “shortcuts” to producing a full dictionary of prime alternating knots. Keywords: alternating knots, knot enumeration, flypes.
1. Knot Universes One goal of the topological theory of knots is to make a dictionary of knots. Since there are infinitely many knots, the hope is to develop an algorithm that can efficiently list all prime knots up to some arbitrary level of complexity. For now I concentrate on enumerating alternating prime knots up to mirror image. A general position projection p : S3 → S2 captures nearly all the information of an alternating knot K ⊂ S3 , so this is our starting point. Definition 1. A (spherical) knot universe of a knot K is the image p(K) of a general position projection p : S3 → S2 (see Figure 1). A crossing of p(K) is a point c ∈ p(K) with two preimage points in p−1 {c}. An m-tangle in a knot universe p(K) is the intersection p(K) ∩ D of the universe with a disc D ⊂ S2 for which ∂D ∩ p(K) consists of m points, none of them crossings. Synonymously, the union −1 of m (T ) ∩ K is also called an m-tangle in K. Customarily, 4-tangles are 2 arcs p simply called tangles. Finally, a knot universe is prime if and only if it contains at least two crossings and all of its 2-tangles are trivial (they contain either all or none of the crossings of the universe). A knot universe p(K) ⊂ S2 , together with a correct assignment of “over” and “under” at each crossing, will completely determine the knot type of K. By picking “overs” and “unders” in a possibly different fashion, we can always determine the knot type of a unique alternating Kalt (up to mirror image) with the same universe. 785
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Figure 1. Knot universe viewed on a section of the sphere. Suppose there is an automorphism of S2 throwing p(K) onto another universe p(K 0 ). Then an appropriate choice of “over” and “under” at each crossing of p(K 0 ) will determine the same alternating knot type as Kalt . We will consider these universes as the same. Definition 2. Suppose c is a crossing in a universe p(K). A flype untwisting c is an ambient isotopy Φ of S3 such that there exist disjoint 3-balls A, B ⊂ S3 and (i) K ∩ A and K ∩ B are tangles in K, (ii) p(K − (A ∪ B)) contains c and no other crossing of p(K), (iii) Φ0 = idS3 , (iv) Φt | A = idA for all 0 ≤ t ≤ 1, (v) Φt | B is a half-turn rotation of B, (vi) p ◦ Φ1 (K − (A ∪ B)) contains exactly one crossing c0 . In this case, we call B the rotating ball for Φ.
Figure 2. Schematic picture of a flype Φ. Notice that a sequence of flypes can change an alternating knot’s universe significantly. At the end of the nineteenth century, Peter Tait conjectured that two least-crossing universes of the same alternating knot will only differ in this way. The Tait Flyping Theorem was finally proven in 1990 by Menasco and Thistlethwaite [4, 5]. Their result provides a correspondence between prime alternating knots and prime spherical knot universes such that: (i) for every knot universe there is a unique alternating knot (up to mirror image), (ii) any two universes corresponding to the same alternating knot are related by a sequence of flypes. In particular, the Flyping Theorem allows us to concentrate only on the enumeration of prime universes.
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2. Flype Structure Let B be the rotating ball for a flype Φ. Since Φ only changes the universe p(K) by untwisting some crossing c ∈ p(K) and creating a new crossing c0 ∈ p(K)Φ on “the other side” of the rotated copy of p(K ∩ B), we customarily identify the two crossings and say that Φ moves c from one location on p(K) to another. Of course, if p(K ∩ B) contains a single crossing, the effects of Φ are unnoticeable. We will only consider the movement, through flyping, of crossings around larger tangles of p(K). Suppose there is a properly embedded disc (D, ∂D) ⊂ (B, ∂B) which is transverse to the axis of rotation of B, intersects K twice, and is disjoint from the preimages p−1 {ci } of the crossings of p(K) (see Figure 3). Rotating each component of B − D by a half-turn in sequence produces the same effect as Φ, so we can view Φ as a sequence of flypes with smaller rotating balls. The tangles captured inside the smallest, non-trivial rotating balls will turn out to be the building blocks of the entire flype structure of a knot.
Figure 3. Rotating B is equivalent to rotating each component of B − D in sequence. Definition 3. Suppose the flype Φ with rotating ball B is not a sequence of flypes with smaller rotating balls. Then p(K∩B) is a flype tangle if and only if it contains at least one crossing of p(K). By performing repeated flypes, a crossing c can be moved between flype tangles beaded around the parallel boundaries of two component discs of S2 − p(K) (see Figure 4). The collection of flype tangles met by c in this circular path is called a flype orbit generated by c. A flype orbit is trivial if it contains less than two flype tangles.
Figure 4. Flyping moves a crossing c around a circular path between flype tangles. The following lemma allows us to unambiguously refer to “the” flype orbit generated by a crossing of p(K).
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Lemma 4. In a prime knot universe, each crossing generates at most one non-trivial flype orbit. Proof. Let c be a crossing in a prime knot universe p(K). There are two ways of untwisting c (see Figure 5) and each way determines a flype orbit.
Figure 5. The two ways of untwisting a crossing. Suppose both flype orbits, A and B, generated by c are non-trivial. Consider the way one set of flype tangles intersects the other.
Figure 6. Two flype orbits, A (solid lines) and B (dotted lines), generated by the same crossing. Each pair of successive B-tangles is connected by two (possibly braided) arcs. Both arcs must lie in the same A-tangle, or else there would exist 3-tangles (see Figure 7 (a)). Furthermore, the B-tangles cannot separate an A-tangle, for otherwise the A-tangle would contain no crossings at all since p(K) is prime (see Figure 7 (b)).
Figure 7. Both (a) 3-tangles and (b) trivial tangles contradict our definitions. Hence there is either only one A-tangle (containing all the braids between Btangles), or only one B-tangle (requiring no such braids).
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If two tangles A ∩ p(K), B ∩ p(K) contain exactly the same set of crossings, we can adjust their boundary so they coincide. We will consider such tangles to be the same. Otherwise we will assume ∂A, ∂B intersect transversely and minimally. Thus, if A ∩ p(K), B ∩ p(K) contain no crossings in common, we can choose A and B to be disjoint. On the other hand, if all the crossings in A ∩ p(K) lie in B ∩ p(K), we can assume the tangles nest and A ⊂ B. The following result states that two flype tangles are either identical, disjoint, nested, or else A ∪ B = S2 with A ∩ B an annulus. Lemma 5. Suppose p(K) is a prime knot universe. If A ∩ p(K), B ∩ p(K) are distinct flype tangles in non-trivial flype orbits, and intersect non-trivially but do not nest, then ∂A ⊂ B and ∂B ⊂ A. Proof. Consider how many points of ∂A ∩ p(K) lie in B and how many points of ∂B ∩ p(K) lie in A. Since ∂(A ∩ B) = (∂A ∩ B) ∪ (∂B ∩ A) is a union of circles, |∂A ∩ B ∩ p(K)| ≡ |∂B ∩ A ∩ p(K)| (mod 2). Now, arbitrary non-trivial tangles A ∩ p(K), B ∩ p(K) cannot intersect minimally in p(K) prime if either (i) |∂A ∩ B ∩ p(K)| and |∂B ∩ A ∩ p(K)| are odd,
Figure 8. Shading indicates intersection-reducing isotopies. (ii) |∂A ∩ B ∩ p(K)| = 0 or |∂B ∩ A ∩ p(K)| = 0,
Figure 9. Shading indicates either intersection-reducing isotopies or a flype tangle containing no crossings. (iii) |∂A ∩ B ∩ p(K)| = 2 and |∂B ∩ A ∩ p(K)| = 4, or |∂A ∩ B ∩ p(K)| = 4 and |∂B ∩ A ∩ p(K)| = 2 (else either A or B contains all crossings of p(K) ).
Figure 10. The shaded tangle contains all the crossings in the universe.
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Suppose that |∂A ∩ B ∩ p(K)| = |∂B ∩ A ∩ p(K)| = 2. Figure 11 shows the six configurations in which ∂B can intersect A. If ∂B intersects A as in (a), then A ∩ p(K) is not a flype tangle, but a union of flype tangles as in Figure 3. Furthermore, the intersections in (b), (d), (e), and (f) are not minimal since there are intersectionreducing isotopies of A and B. Hence the only possibility remaining is that ∂B ∩ A is as in (c).
Figure 11. ∂B can intersect A in six ways when |∂A∩B ∩p(K)| = |∂B ∩ A ∩ p(K)| = 2 . If next to A, there were a crossing c which generated its flype orbit, ∂B could be adjusted so that |∂B∩A∩p(K)| = 3 while keeping |∂A∩∂B| constant (see Figure 12); but then ∂B could be further adjusted to reduce ∂A ∩ ∂B as in Figure 8.
Figure 12. ∂A ∩ ∂B may be reduced by first isotopying as shaded. Thus on each side of A lie flype tangles A0 ∩ p(K), A00 ∩ p(K) in its flype orbit. Arguing as above, ∂B ∩ p(K) must have more than one point in each of A0 , A00 . Therefore A0 and A00 must in fact be the same, since ∂B ∩ p(K) has only two points outside A. However, this flype orbit is not generated by any crossing (see Figure 13), contrary to our definitions.
Figure 13. This flype orbit is not generated by any crossing. Thus |∂A ∩ B ∩ p(K)| = |∂B ∩ A ∩ p(K)| = 4, and hence ∂A ⊂ B and ∂B ⊂ A as required. Corollary 6. Suppose A, B are flype orbits in a prime universe p(K) with B generated by a crossing in the A-tangle A ∩ p(K). Then ∂A can be adjusted so each B-tangle either lies completely in A or contains all crossings of p(K) − A.
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Proof. The lemma above asserts that flype tangles intersecting A ∩ p(K) nontrivially will either lie in A or contain S2 − A as desired. Suppose no single B-tangle contains all crossings in p(K) − A; then every B-tangle that intersects A lies completely in A. A slight modification on the proof of Lemma 4 shows that there is only one B-tangle outside A (see Figure 14).
Figure 14. Much as in Figure 6, the way in which A-tangles intersect B-tangles allows for only one B-tangle outside A. This B-tangle must contain all crossings in p(K) − A.
Flypes act on the set of topologically distinct knot universes corresponding to an alternating knot K by moving crossings across flype tangles. Suppose A and B are distinct flype orbits in p(K). Then the action of any flype Φ which moves crossings across B-tangles is independent from the action of any flype moving crossings across A-tangles. For suppose Φ moves a crossing c. Since A = 6 B, c does not generate A and must lie in some A-tangle A ∩ p(K). By the corollary above, every B-tangle is either completely inside A or contains all crossings of S2 − A. In particular only one such tangle does the latter, and since every movement of c can be achieved by flypes that hold this tangle fixed, the action of Φ affects p(K) only inside A. However, flypes moving crossings across A-tangles affect p(K) only outside A. Thus the action of flypes on p(K) may change flype tangles but preserves the orbit structure (the number of flype orbits, the number of tangles in each orbit, and the crossings contained in each tangle). By performing flypes, we can move the crossings that generate a flype orbit A until they lie together in a two-arc braid. Further flyping allows us to move this amalgam of crossings between flype tangles in the orbit. Since flype actions affect flype orbits independently, all crossings generating each flype orbit may be grouped in this way. We can then classify the action of flypes to these special knot universes. Definition 7. A knot universe is in flype-minimal position if all crossings that generate a single flype orbit are grouped together in a two-arc braid. A flype is minimal if it moves an entire grouping of generator crossings across a single flype tangle. For a prime alternating knot K, let U(K) denote the graph having the collection of flype-minimal position knot universes corresponding to K as vertices and whose edges indicate the action of minimal flypes. For example, consider the knot K pictured in Figure 15. There are two non-trivial flype orbits in K, each containing a 10-crossing flype tangle and three 3-crossing flype tangles. Hence we can perform 16 distinct and minimal flypes (including the identity) as indicated by the torus lattice in Figure 16. However, there are only
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Figure 15. An alternating knot K. three distinct prime universes for K (labelled A, B, and C), since the large amount of symmetry in K induces automorphisms of S2 fixing the universes of K. For instance, consider the four universes in Figure 16 labelled A. The top left universe is identified to the bottom left universe by a reflection across the horizontal equator of S2 = R2 ∪{∞}, and to the bottom right universe by a rotation of R2 by π followed by a translation of ∞. Analogous automorphisms identify the top right universe with the bottom ones, and thus they all correspond to the same point in U(K).
Figure 16. A schematic diagram of the action of flypes on the prime universes of K = (1, 3, 3, 3, (1, 3, 3, 3)). Corollary 8. The space U(K) of flype-minimal position knot universes of a prime alternating knot K with f non-trivial flype orbits has the structure of a quotient of an f -dimensional torus lattice.
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Proof. By the Tait Flyping Theorem, all universes corresponding to K are related through flypes; in particular, one flype-minimal position universe can be transformed into another via minimal flypes. Therefore U(K) is connected. The action of flypes that move each set of generator crossings is cyclic, since the flype tangles in each orbit lie on the parallel boundary of two discs in S2 − p(K). The actions of flypes moving different amalgams of generator crossings are independent, and the order in which these flypes are performed is irrelevant. U(K) is hence structured by this action, forming an f -dimensional torus lattice. However, self-symmetries of K may identify vertices in this lattice (as in Figure 16 above), making U(K) a quotient of the f -dimensional torus lattice.
3. Twisted Splices By cutting, twisting, and re-glueing, we can build complicated prime alternating knots out of simpler ones. Much like the aufbau principle of chemistry, we can start at the bottom of knot complexity, the unknot, and build up new knots by adding twists. 1 [2] shows this method of building up knots provides an enumeration algorithm, which we first implemented in [3]. Here I present a new proof and some insights that may help optimize the process. Definition 9. A twisted splice on a knot universe p(K) is the replacement χ of a trivial tangle with a two-arc braided tangle resulting in a knot universe p(K)χ , as in Figure 17.
Figure 17. Local picture of a twisted splice; depending on the orientation of the trivial tangle, either an even or an odd number of crossings may be added. In particular, if p(K) were oriented, the number of crossings in the new tangle must be even when the original arcs flow in the same direction, and odd when they flow in opposite directions. A crossing c ∈ p(K)χ is a new twist on p(K) if and only if it forms part of the new tangle in p(K)χ . Unless p(K) is trivial, we will assume the arcs in the trivial tangle to be spliced are contained in distinct components of p(K) − {crossings} . This is a technical point, but one necessary to insure that the results of twisted splicing are prime. 1It is noteworthy that connections between knot theory and chemistry date back to some of
the earliest endeavors in knot enumeration, spearheaded by Lord Kelvin’s hypothesis that atoms were knotted vortices in the fabric of the ether (see [1], pp. 5, 31).
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Lemma 10. If the knot universe p(K) is either prime or trivial (it contains no crossings), then the knot universe p(K)χ is prime. Proof. If p(K) is trivial, then p(K)χ will be the universe of a (2, n)-torus knot (with n odd), which are always prime. Suppose, then, that p(K) is prime but p(K)χ is not. Let A ∩ p(K)χ be the new tangle in p(K)χ , B = S2 − A, and C ∩ p(K)χ be a non-trivial 2-tangle, chosen so ∂C intersects ∂A transversely. There is an exact copy of B ∩ p(K)χ in p(K), which is prime, so C cannot lie entirely in B. In particular, ∂C ∩ A ∩ p(K)χ 6= ∅. As a two-arc braid, A ∩ p(K)χ cannot be intersected by ∂C other than cross-sectionally (see Figure 18).
Figure 18. A non-trivial 2-tangle in p(K)χ indicates that p(K) is a split link. Thus, both points of ∂C∩p(K)χ lie in A and ∂C∩B misses p(K)χ altogether. In that case, there is a circle in S2 which separates p(K) as a split link, a contradiction. Twisted splicing will change the orientation of half of p(K), and so p(K)χ inherits no unique orientation from p(K). However, regardless of the orientation given to p(K)χ , it is always the case that both arcs of the new tangle will flow in the same direction. In particular, any orientation on p(K)χ will indicate how to snip, untwist, and rejoin arcs of p(K)χ to recover p(K). This observation gives us a remarkable connection between the set of crossings that generate only trivial flype orbits in a prime universe and the set of new twists. Lemma 11. Every crossing that only generates trivial flype orbits in a prime knot universe p(K) is a new twist on some prime or trivial universe p(K 0 ) such that p(K) = p(K 0 )χ . Proof. Let c ∈ p(K) generate only trivial flype orbits. Performing the adequate snipping, untwisting, and rejoining of arcs in p(K) to eliminate c will produce a knot universe p(K 0 ). Suppose that p(K 0 ) is neither prime nor trivial. Let A ∩ p(K 0 ) be the trivial tangle removed to make p(K) = p(K 0 )χ , B = S2 − A, and C ∩ p(K 0 ) be a non-trivial 2-tangle chosen so ∂C intersects ∂A transversely. There is an exact copy of B ∩ p(K 0 ) in p(K), which is prime, so C cannot lie entirely in B. In particular, ∂C must separate the two arcs of A ∩ p(K 0 ), intersecting p(K 0 ) twice inside B. By our construction of p(K 0 ), both C and S2 − C must each contain more than a single crossing of p(K 0 ). Hence c generates a non-trivial flype orbit in p(K) (see Figure 19), contradicting our choice of c. Thus p(K 0 ) is prime.
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Figure 19. A non-trivial 2-tangle in p(K) indicates that c ∈ p(K)χ generates a non-trivial flype orbit. Lemma 12. In a prime knot universe p(K), there is always some crossing which generates only trivial flype orbits. Proof. Let a0 be a crossing in p(K); if it generates only trivial flype orbits, a0 is the desired crossing. Suppose, then that a0 generates a non-trivial flype orbit A0 . Let a1 be a crossing in some tangle (A0 ∩ p(K)) ∈ A0 ; a1 is the desired crossing if it generates only trivial flype orbits, so suppose that it generates a non-trivial flype orbit A1 . By Corollary 6, all but one of the tangles in A1 lie entirely in A0 , so let a2 be a crossing in some tangle (A1 ∩ p(K)) ∈ A1 with A1 ⊂ A0 . We can continue inductively, choosing crossings a0 , . . . , an until an generates only trivial flype orbits; otherwise, let An be the non-trivial flype orbit generated by an , and pick a crossing an+1 in some flype tangle (An ∩ p(K)) ⊂ An with An ⊂ An−1 . Since An contains strictly fewer crossings than An−1 and A0 contains a finite number of crossings, this process must eventually terminate with a crossing an which generates only trivial flype orbits. Note that the choice of a0 and A0 in the proof of Lemma 12 is arbitrary. Given any flype tangle B ∩ p(K), we may pick both a0 ∈ B ∩ p(K) and (if necessary) A0 ⊂ B to guarantee an ∈ B ∩ p(K). This proves the following special case. Corollary 13. In every flype tangle in a prime universe p(K), there is some crossing which generates only trivial flype orbits. Lemmas 11, 12, and Corollary 13 allow us to prove the main theorem of [2], as well as some improvements. Theorem 14. Every prime knot universe is the result of a sequence of twisted splices on the trivial universe, and the result of each step in this sequence is prime. Proof. Since p(K) is prime, it contains some crossing c generating only trivial flype orbits; thus c is a new twist on a prime or trivial universe p(K 0 ) with p(K) = p(K 0 )χ0 . Since p(K 0 ) contains fewer crossings than p(K), induction gives a sequence of twisted splices χ1 , χ2 , . . . , χm−1 , χm on the trivial universe p(U ) such that p(K) = p(U )χ1 χ2 ···χm−1 χm χ0 . Furthermore, each step in the sequence is prime by Lemma 10.
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Corollary 15. The sequence of twisted splices producing a prime knot universe from the trivial universe can be chosen so that the flype tangles in any flype orbit are constructed one at a time. Proof. Order the flype tangles in the flype orbit A in a prime knot universe p(K) as A = {A1 ∩ p(K), . . . , An ∩ p(K)}. Suppose inductively that prime universes with fewer crossings than p(K) have a tangle-by-tangle construction, taking the universe of the trefoil knot as the base case. Corollary 13 gives a crossing c ∈ An ∩ p(K) which generates only trivial flype orbits; then c is a new twist on a prime or trivial universe p(K 0 ) with p(K) = p(K 0 )χ0 . Since twisted splicing is a local change, p(K 0 ) has an exact copy of p(K) − An ; in place of An ∩ p(K), p(K 0 ) has a (possibly empty) collection of flype tangles B1 ∩ p(K 0 ), . . . , Bn0 ∩ p(K 0 ) connected by two-arc braids so that {A1 ∩ p(K), . . . , An−1 ∩ p(K), B1 ∩ p(K 0 ), . . . , Bn0 ∩ p(K 0 )} is a flype orbit for p(K 0 ). Then induction gives a sequence of twisted splices χ1 , . . . , χm on the trivial universe p(U ) and a sequence of integers 1 < q1 < · · · < qn−1 ≤ m, such that p(K) = p(K 0 )χ0 = p(U )χ1 ···χm χ0 , and for each i < n, p(U )χ1 ···χqi consists of an exact copy of (A1 ∪ · · · ∪ Ai ) ∩ p(K) plus a two-arc braid in place of each of Ai+1 ∩ p(K), . . . , An ∩ p(K). Then this is the desired sequence of splices. Corollary 16. The sequence of twisted splices producing a prime knot universe from the trivial universe can be chosen so that each splice adds no more crossings than the very first splice. Proof. The universe of a (2, n)-torus knot p(K1 ) will be called an ancestor of p(K2 ) if there is a sequence of twisted splices χ1 , . . . , χm on the trivial universe p(U ) such that p(K1 ) = p(U )χ1 and p(K2 ) = p(U )χ1 ···χm . Call p(K1 ) the largest ancestor of p(K2 ) if p(K1 ) has more crossings than any other ancestor of p(K2 ). Suppose, then, that p(K) is the least-crossing prime knot universe with the property that for every sequence of twisted splices χ1 , . . . , χm on the trivial universe p(U ) for which p(K) = p(U )χ1 ···χm and p(U )χ1 is the largest ancestor of p(K), there is some χi adding more crossings than χ1 . Let p(K 0 ) be p(K)’s largest ancestor, say with t crossings. Let χ1 , . . . , χm be any sequence of twisted splices on p(U ) with p(K 0 ) = p(U )χ1 ,
p(K) = p(U )χ1 ···χm .
Since p(U )χ1 ···χm−1 has fewer crossings than p(K) but the same largest ancestor, we can choose χ1 , . . . , χm so that for each i < m, χi adds no more than t crossings. Thus χm must add at least t + 1 crossings. Together, these crossings generate a flype orbit with at least one flype tangle. Corollary 15 gives a sequence of twisted splices χ01 , . . . , χ0m0 producing p(K) from p(U ) which builds this flype orbit one tangle at a time. Thus p(U )χ1 is the universe of a torus knot with at least t + 1 crossings. But then p(U )χ1 is an ancestor of p(K) with more crossings than p(K 0 ), a contradiction.
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4. Enumeration Theorem 14 and Corollaries 15 and 16 indicate the path that our aufbau process must follow. Launching inductively from the unknot, we build all n-crossing prime knot universes from previously generated universes by performing twisted splices between arcs on the boundary of the discs of S2 − p(K). Thus, the unknot gives rise to the trefoil universe (31 ) , which then yields the figure-eight knot universe (41 ). As the aufbau algorithm continues, the (2, 5)-torus knot universe (51 ) emerges from the unknot, while splices in either the trefoil or the figure-eight knot produce the universe for knot 52 . By keeping a record of each universe’s flype structure, we could restrict the splicing to only certain areas, building flype orbits one tangle at a time as in Corollary 15. If we also carry the number of crossings of each universe’s largest ancestor as an extra datum in the enumeration process, we can further limit the number of twisted splicing performed. The amount of information required to limit ourselves to sequences of splices as in Corollary 15 seems staggering at this point. Perhaps it is more advantageous to store only largest ancestor information with each newly generated “seed” universe. Then to produce new n-crossing universes from s-crossing seeds with a t-crossing largest ancestor, all possible twisted splices would be executed as long as n − s ≤ t. Finally, we exploit Corollary 8 to recover the information about the alternating knot types associated to the obtained universes. According to the Tait Flyping Theorem, any two least-crossing knot universes of an alternating knot K are flypeequivalent. In particular, K has a flype-minimal position universe. To choose a single representative among all universes representing K, our algorithm may as well discard those universes not in flype-minimal position. This leaves only the universes associated to vertices of U(K), which may be examined by circling through the torus lattice underlying U(K). In this fashion, the removal of flype-equivalent universes renders the desired dictionary of alternating knots. Acknowledgements First of all, I would like to dedicate this paper to my mother. In addition, my many thanks go to Prof. Kenneth Millett, whose advice at every level of this investigation proved invaluable. This research was funded in part by a National Science Foundation Graduate Research Fellowship.
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References [1] Adams, C. C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W. H. Freeman and Co., New York (1994). [2] Arnold, E., C. E. Fan, S. Pierre, C. C. Torres, Towards an Enumeration of Knots, unpublished research from the Summer Academic Research Internship at U.C. Santa Barbara (1990). [3] Calvo, J. A., J. Bandera, J. Golingo, S. Talton, Implementing an Enumeration of Prime Knots, unpublished research from the Summer Academic Research Internship at U.C. Santa Barbara (1991). [4] Menasco, W. W., M. B. Thistlethwaite, The Tait Flyping Conjecture, Bulletin (New Series) of the American Mathematical Society 25 (1991), no. 2, 403 – 412. [5] , The Classification of Alternating Links, Annals of Mathematics 138 (1993), 113 – 171.
