arXiv:0704.1941v1  [math.GT]  16 Apr 2007
TAIT’S CONJECTURES AND
ODD CROSSING NUMBER AMPHICHEIRAL KNOTS
A. Stoimenow∗
Research Institute for Mathematical Sciences,
Kyoto University, Kyoto 606-8502, Japan
e-mail: stoimeno@kurims.kyoto-u.ac.jp
WWW: http://www.kurims.kyoto-u.ac.jp/˜stoimeno/
Abstract. We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his
pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We
announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last re-
maining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An
application to the non-triviality problem for the Jones polynomial is also outlined.
Keywords: Jones polynomial, amphicheiral knot, crossing number
AMS subject classification: 57M25 (primary), 01A55, 01A60 (secondary)
1. The first knot tables
Knot theory took its origins in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and
J. Maxwell propagated the vortex-atom theory, in an attempt to explain the structure of the universe. They believed
that a super-substance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby
understood as tying a piece of rope, and then identifying its both ends so that the tying cannot be any more undone.
Thus, in the realm of constructing a periodic table of elements, Tait began the catalogisation of the “simplest”
knots. He depicted knots (as we still do today) by diagrams, consisting of a (smooth) plane curve with transverse
self-intersections, or crossings. At each crossing one of the two strands passes over the other. The above notion
of simplicity refers to the number of crossings of the diagram. Tait’s list aimed at presenting, among (knots with)
diagrams of few crossings, each knot by exactly one diagram. Alternatively we can define the crossing number of
a knot as the minimal crossing number of all its diagrams, and say that we seek the list of knots with given (small)
crossing number. We also like knots represented by different diagrams in the list to be inequivalent, in the sense
that one cannot turn a (closed) piece of rope knotted the one way into one knotted the other way, without cutting
the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or
unknot. It has some special importance, much like the unit element in a group.
Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued
his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even
though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is given, with emphasis
on its history, in [S], and from a more contemporary point of view in [HTW, H].
2. Tait’s conjectures
Tait’s accomplishment allows us to call him with some right the first knot theorist. Yet Tait worked mainly by
intuition. He had at his time no rigorous way of showing knots inequivalent. Tait’s reasoning is not easy to interpret
precisely nowadays, had it been formulated in a language quite different from (and far less developed than) our
present. Nonetheless he evidently observed several phenomena, which, apart from knot tabulation, would become
a legacy to his successors.
This is a preprint. I would be grateful for any comments and corrections. Current version: November 2, 2018 First version: April 13, 2007
∗Financial support by the 21st Century COE Program is acknowledged.
http://arxiv.org/abs/0704.1941v1
2 2 Tait’s conjectures
unknot left-hand trefoil right-hand trefoil figure-8 knot
Figure 1
It seems to remain unclear whether Tait was convinced certain properties to hold for all, or just for alternating knots.
A knot is alternating, if in some (alternating) diagram the curve passes crossings interchangingly over-under like
, i.e. not containing or . The knots in figure 1 are such. In fact, this is true for all knots
up to 7 crossings, catalogued by Tait, and at least for a large portion of the slightly more complicated ones he was
shown by his successors in his lifetime (even though it is known now that alternation is a rare property for generic
crossing numbers [Th3]). Thus, certainly Tait was guided by evidence from alternating diagrams. Their occurrence
in the tables suggested to him
Conjecture 2.1 (Tait’s conjecture I) A reduced, i.e. not of the form , alternating diagram has min-
imal crossing number (for the knot it represents).
For the next problem, we need to define the writhe. If one equips (the curve of) a knot diagram with an orientation,
then each crossing looks, if observed from an appropriate angle, locally like (positive crossing) or (negative
crossing). The writhe is the difference between the number of former crossings and the number of latter. (An easy
observation shows that the writhe is the same for either orientation.)
Conjecture 2.2 (Tait’s conjecture II) Minimal crossing number diagrams of the same (alternating?) knot have the
same writhe.
For alternating diagrams, he conjectured more precisely the following:
Conjecture 2.3 (Tait’s conjecture III) Alternating diagrams of the same knot are related by a sequence of flypes:
PQ ←→
One can easily observe that a flype preserves the writhe, and so conjecture III implies conjecture II (for alternating
knots1).
For Tait’s last problem, we consider an amphicheiral knot. Such a knot can be turned into its mirror image. From
the knots in figure 1, the unknot is obviously amphicheiral. So is the figure-eight knot, as shows a simple exercise.
In contrast, the trefoil is not amphicheiral. In other words, the left-hand trefoil and its mirror image, the right-hand
trefoil, are two distinct knots (a fact that stubbed knot theorists for a while, and was first proved by Max Dehn). Tait
was wondering what crossing numbers amphicheiral knots can have. The evidence he had in mind can probably be
formulated so:
1up to some technical issues of primeness and whether all minimal crossing diagrams are alternating. These issues are settled, but we like to
skip them here for simplicity.
Conjecture 2.4 (Tait’s conjecture IV) Amphicheiral (alternating?) knots have even crossing number.
Note that (for alternating knots, and with the remark in footnote 1) this is a consequence of conjecture II, for
mirroring a diagram interchanges positive and negative crossings, and so negates the writhe.
3. Reidemeister moves and invariants
A few decades after their genesis, Tait’s knot lists would be proved right. With the work of Alexander, Reidemeister
and others knot theory began to be put on a mathematical fundament. Reidemeister showed that three types of local
moves (i.e. moves altering only a fragment of the diagrams, as shown below) suffice to interrelate all diagrams of
the same knot.
←→ ←→ ←→ ←→ ←→ ←→
To distinguish two knots thus translates into the question how to prove that two diagrams (of these knots) are not
connected by a sequence of such moves. This is done with the help of an invariant, that is, a map
{ knot diagrams }→ “something” ,
whose value in “something” does not change (is invariant) when the argument (diagram) is changed by a Reide-
meister move. The question what “something” should be is justified. The answer is that it would suffice to be any
set of objects whose distinctness is easy to verify, yet which is large enough to allow the invariant to take many
different values. For us it will be the ring Z[t, t−1] of Laurent polynomials in one variable with integer coefficients.
Clearly one can compare coefficients easier than wondering about a sequence of Reidemeister moves, which may
be arbitrarily long and pass over arbitrarily complicated intermediate diagrams. (There is a fundamental, but out
of our focus here, issue how to, and that one can to some extent, control these sequences [HL].) An obvious other
desirable feature of an invariant is that we could easily evaluate it from a diagram.
Alexander’s merit was to construct precisely such an invariant. The Alexander polynomial [Al] remained (and still
remains) a main theme in knot theory for decades to come. Let us observe, though, that we actually already came
across one other knot invariant, the crossing number. It is an invariant directly by definition, since it was defined on
the whole Reidemeister move equivalence class of diagrams. However, this definition makes it difficult to evaluate
from a diagram – in contrast to Alexander’s polynomial, for which its creator, and later many others, gave several
simple procedures.
4. The Jones polynomial
60 years after Alexander, a new chapter of knot theory was opened by V. Jones, with the discovery of a successor
to Alexander’s polynomial. The developments the Jones polynomial V [J] has sparked in the 20 years since its
appearance are impossible even to be vaguely sketched in completeness, and go far beyond both the competence
and expository intention of the author here. Several more concepts, like links, braids, geometric, Vassiliev and
quantum invariants etc., are left out for simplicity and length reasons, for which the author likes to apologise at this
point. A good (though still partial, and now no longer very recent) account on these issues was given by Birman
[Bi].
Let us recall, though, one of the first achievements the Jones polynomial became famous with – the solution of Tait’s
conjectures. Conjectures I, II and IV, for alternating knots, were proved by Kauffman [Kf2], Murasugi [Mu, Mu2]
and Thistlethwaite [Th2]. We will need a few more words on Kauffman’s proof, since it uses a calculation procedure
for V called state model. This state model also gives a very elementary proof that the Jones polynomial is an
invariant. (Kauffman had previously developed a similar model for the Alexander polynomial, too [Kf3].)
Based on Kauffman’s state model, Lickorish and Thistlethwaite [LT] defined a (semi)adequate knot and diagram.
An advantage of this concept is that alternating diagrams/knots are adequate. Many details would better be skipped
here, but let us say that a diagram is adequate/semiadequate if it is +adequate and/or−adequate, and that taking the
mirror image of a diagram transforms the property for + into the one for −. A knot with some of these properties
4 6 Semiadequacy invariants and the non-triviality problem
is defined as one that has a diagram with the feature of the same name. Thistlethwaite extended the proof of the
Tait’s three conjectures to the class of adequate knots [Th], applying a 2-variable generalisation of V , the Kauffman
polynomial [Kf].
For a general (non-alternating) knot, in case of conjecture II, Tait’s intuition had been proved misleading. In
the 1970’s, K. Perko [Pe] observed that there are two 10 crossing knots in the tables [Ro, appendix], which are
equivalent, even though their 10 crossing diagrams have different writhe. This duplication had remained unnoticed
for quite a while, possibly due to the belief in Tait’s conjecture. Remedying this error (and a few other long-
remained ones) in the tables still causes some confusion in their use.
Tait’s conjecture III was settled a few years after the others by Menasco and Thistlethwaite [MT], mostly using
geometric techniques, though again with some (now subordinate) appearance of the Jones polynomial.
5. The crossing numbers of amphicheiral knots
Kauffman, Murasugi and Thistlethwaite’s proof of Tait’s conjecture IV shows that an alternating knot K of odd
crossing number and its mirror image !K always have distinct Jones polynomials. (Let us in contrast remark that K
and !K have the same Alexander polynomial for every knot K.) In the opposite direction, their work allows to easily
find a(n alternating) amphicheiral knot of every even crossing number at least 4. But their results could not decide
what crossing number non-alternating amphicheiral knots can have. The main aim of this note is to announce the
complete solution to Tait’s (last?) problem.
Theorem 5.1 For each odd natural number n≥ 15, there exists an amphicheiral knot of crossing number n.
Similarly to Perko’s knot, a particular instance disproving Tait’s conjecture IV for non-alternating knots was found
accidentally: Hoste and Thistlethwaite, in the course of routine knot tabulation, discovered an amphicheiral 15
crossing knot. (Their compilational work had previously shown that there are no amphicheiral knots of odd crossing
numbers up to 13.) Settling the other crossing numbers is a major problem, though, since exhaustive enumeration
is no longer a feasible approach – we face the above noticed difficulty that we do not know (generally) how to
determine the crossing number. A few other methods are known, but all they fail on such examples. Thus the
way to our result is rather far, and below we will conclude by just giving a brief outline of the proof. The details
will appear in a separate (long) paper. We also mention another application of our approach, which addresses the
non-triviality problem for the Jones polynomial.
6. Semiadequacy invariants and the non-triviality problem
The Alexander polynomial was, from its very beginning, connected to topological features of knots. The situation
is rather different for its successor. The problem to give a topological meaning to the Jones polynomial has bothered
many knot theorists ever since this invariant appeared. So far we still find ourselves in the embarrassing state where
we “can quickly fill pages with the coefficients and exponents of V for not-too-complicated knots without having
the slightest idea what they mean” ([Bi, end of §3]). Similarly unsolved, and intriguing, remains the problem,
formulated by Jones, if his polynomial detects the unknot. Again, “our lack of knowledge about this problem
is in striking contrast to the control mathematicians now have over the Alexander polynomial: understanding its
topological meaning, we also know precisely how to construct knots with Alexander polynomial 1” (ibid., rem. (iii)
after theorem 3, §8; see, though, also [EKT]).
In an attempt to gain more insight into the appearance of the Jones polynomial, the author, and (up to minor
interaction, independently) Dasbach and Lin [DL, DL2], initiated a detailed study of some coefficients of V in
semiadequate diagrams. Let us remark here that, while adequacy is only a slight extension of alternation, semiad-
equacy is a rather wide extension of adequacy. (For the experts: semiadequate knots contain completely positive,
Montesinos and 3-braid knots.) Semiadequacy is still a fairly general condition, yet it helps settle many technical
issues.
For semiadequate knots the first coefficient of the Jones polynomial is ±1, almost by definition [LT]. The outcome
of our work was that we gained an understanding of coefficients 2 and 3. Their invariance allows to derive 3
invariant quantities each from a +adequate, and similarly from a −adequate diagram, called below semiadequacy
References 5
invariants. Their merit is that they reflect directly certain features of the diagram, and so we have a precise idea
how a semiadequate diagram with given invariants must look like. The first of them allows to prove:
Theorem 6.1 Semiadequate knots have non-trivial Jones polynomial.
This implies (for experts) the result also for Montesinos and 3-braid knots, but it can be proved also for their
Whitehead doubles, some strongly n-trivial knots and k-almost positive knots with k ≤ 3.
Our three semiadequacy invariants become also, joined by a relative obtained from the Kauffman polynomial and
Thistlethwaite’s results [Th], the main tool for the proof of theorem 5.1. For given odd n ≥ 15, we start with
an amphicheiral knot K that has an n crossing diagram, which is semiadequate. Luckily, such examples can be
obtained by leaning on Hoste-Thistlethwaite’s knot. The work in [Th] shows then that the crossing number of K
is at least n− 1, and were it n− 1, a minimal crossing diagram D would be adequate. Then we have 4 invariants
for both +adequacy and −adequacy each available. A detailed study of how an n− 1 crossing diagram with such
invariants must look like is necessary to exclude most cases for D. Hereby, among the various generalisations
of Thistlethwaite’s knot, one must choose carefully the one whose invariants make the exclusion argument most
convenient (or better to say, feasible at all). Only a small fraction of possibilities for D remain, which are easy to
check, and rule out, by computer. This allows us to conclude that in fact D cannot exist.
References
[Al] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275–306.
[Bi] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28(2) (1993), 253–287.
[Co] J. H. Conway, On enumeration of knots and links, in “Computational Problems in abstract algebra” (J. Leech, ed.),
Pergamon Press, 1969, 329–358.
[DL] O. Dasbach and X.-S. Lin, A volume-ish theorem for the Jones polynomial of alternating knots, math.GT/0403448,
to appear in Pacific J. Math.
[DL2] ” and ” , On the Head and the Tail of the Colored Jones Polynomial, Compositio Math.
142(5) (2006), 1332–1342.
[EKT] S. Eliahou, L. H. Kauffman and M. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology
42(1) (2003), 155–169.
[HL] J. Hass and J. C. Lagarias, The number of Reidemeister moves needed for unknotting, J. Am. Math. Soc. 14(2) (2001),
399–428.
[H] J. Hoste, The enumeration and classification of knots and links, “Handbook of Knot Theory” (W. Menasco and M.
Thistlethwaite, eds.), Elsevier (2005), 209–232.
[HTW] ” , M. Thistlethwaite and J. Weeks, The first 1,701,936 knots, Math. Intell. 20 (4) (1998), 33–48.
[J] V. F. R. Jones, A polynomial invariant of knots and links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985),
103–111.
[Kf] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417–471.
[Kf2] ” , State models and the Jones polynomial, Topology 26 (1987), 395–407.
[Kf3] ” , Formal knot theory, Mathematical Notes 30, Princeton University Press, Princeton, NJ, 1983.
[LT] W. B. R. Lickorish and M. B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers,
Comment. Math. Helv. 63 (1988), 527–539.
[MT] W. W. Menasco and M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403–412.
[Mu] K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187–194.
[Mu2] ” , Jones polynomials and classical conjectures in knot theory II, Math. Proc. Cambridge Philos. Soc.
102(2) (1987), 317–318.
[Pe] K. A. Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266.
[Ro] D. Rolfsen, Knots and links, Publish or Perish, 1976.
[S] D. S. Silver, Knot Theory’s Odd Origins, American Scientist 94(2) (2006), 158–165.
[Th] M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93(2) (1988), 285–296.
[Th2] ” , A spanning tree expansion of the Jones polynomial, Topology 26(3) (1987), 297–309.
[Th3] ” , On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7(7)
(1998), 981–1004.

  We give a brief historical overview of the Tait conjectures, made 120 years
ago in the course of his pioneering work in tabulating the simplest knots, and
solved a century later using the Jones polynomial. We announce the solution,
again based on a substantial study of the Jones polynomial, of one (possibly
his last remaining?) problem of Tait, with the construction of amphicheiral
knots of almost all odd crossing numbers. An application to the non-triviality
problem for the Jones polynomial is also outlined.

arXiv:0704.1941v1  [math.GT]  16 Apr 2007
TAIT’S CONJECTURES AND
ODD CROSSING NUMBER AMPHICHEIRAL KNOTS
A. Stoimenow∗
Research Institute for Mathematical Sciences,
Kyoto University, Kyoto 606-8502, Japan
e-mail: stoimeno@kurims.kyoto-u.ac.jp
WWW: http://www.kurims.kyoto-u.ac.jp/˜stoimeno/
Abstract. We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his
pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We
announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last re-
maining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An
application to the non-triviality problem for the Jones polynomial is also outlined.
Keywords: Jones polynomial, amphicheiral knot, crossing number
AMS subject classification: 57M25 (primary), 01A55, 01A60 (secondary)
1. The first knot tables
Knot theory took its origins in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and
J. Maxwell propagated the vortex-atom theory, in an attempt to explain the structure of the universe. They believed
that a super-substance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby
understood as tying a piece of rope, and then identifying its both ends so that the tying cannot be any more undone.
Thus, in the realm of constructing a periodic table of elements, Tait began the catalogisation of the “simplest”
knots. He depicted knots (as we still do today) by diagrams, consisting of a (smooth) plane curve with transverse
self-intersections, or crossings. At each crossing one of the two strands passes over the other. The above notion
of simplicity refers to the number of crossings of the diagram. Tait’s list aimed at presenting, among (knots with)
diagrams of few crossings, each knot by exactly one diagram. Alternatively we can define the crossing number of
a knot as the minimal crossing number of all its diagrams, and say that we seek the list of knots with given (small)
crossing number. We also like knots represented by different diagrams in the list to be inequivalent, in the sense
that one cannot turn a (closed) piece of rope knotted the one way into one knotted the other way, without cutting
the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or
unknot. It has some special importance, much like the unit element in a group.
Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued
his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even
though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is given, with emphasis
on its history, in [S], and from a more contemporary point of view in [HTW, H].
2. Tait’s conjectures
Tait’s accomplishment allows us to call him with some right the first knot theorist. Yet Tait worked mainly by
intuition. He had at his time no rigorous way of showing knots inequivalent. Tait’s reasoning is not easy to interpret
precisely nowadays, had it been formulated in a language quite different from (and far less developed than) our
present. Nonetheless he evidently observed several phenomena, which, apart from knot tabulation, would become
a legacy to his successors.
This is a preprint. I would be grateful for any comments and corrections. Current version: November 2, 2018 First version: April 13, 2007
∗Financial support by the 21st Century COE Program is acknowledged.
http://arxiv.org/abs/0704.1941v1
2 2 Tait’s conjectures
unknot left-hand trefoil right-hand trefoil figure-8 knot
Figure 1
It seems to remain unclear whether Tait was convinced certain properties to hold for all, or just for alternating knots.
A knot is alternating, if in some (alternating) diagram the curve passes crossings interchangingly over-under like
, i.e. not containing or . The knots in figure 1 are such. In fact, this is true for all knots
up to 7 crossings, catalogued by Tait, and at least for a large portion of the slightly more complicated ones he was
shown by his successors in his lifetime (even though it is known now that alternation is a rare property for generic
crossing numbers [Th3]). Thus, certainly Tait was guided by evidence from alternating diagrams. Their occurrence
in the tables suggested to him
Conjecture 2.1 (Tait’s conjecture I) A reduced, i.e. not of the form , alternating diagram has min-
imal crossing number (for the knot it represents).
For the next problem, we need to define the writhe. If one equips (the curve of) a knot diagram with an orientation,
then each crossing looks, if observed from an appropriate angle, locally like (positive crossing) or (negative
crossing). The writhe is the difference between the number of former crossings and the number of latter. (An easy
observation shows that the writhe is the same for either orientation.)
Conjecture 2.2 (Tait’s conjecture II) Minimal crossing number diagrams of the same (alternating?) knot have the
same writhe.
For alternating diagrams, he conjectured more precisely the following:
Conjecture 2.3 (Tait’s conjecture III) Alternating diagrams of the same knot are related by a sequence of flypes:
PQ ←→
One can easily observe that a flype preserves the writhe, and so conjecture III implies conjecture II (for alternating
knots1).
For Tait’s last problem, we consider an amphicheiral knot. Such a knot can be turned into its mirror image. From
the knots in figure 1, the unknot is obviously amphicheiral. So is the figure-eight knot, as shows a simple exercise.
In contrast, the trefoil is not amphicheiral. In other words, the left-hand trefoil and its mirror image, the right-hand
trefoil, are two distinct knots (a fact that stubbed knot theorists for a while, and was first proved by Max Dehn). Tait
was wondering what crossing numbers amphicheiral knots can have. The evidence he had in mind can probably be
formulated so:
1up to some technical issues of primeness and whether all minimal crossing diagrams are alternating. These issues are settled, but we like to
skip them here for simplicity.
Conjecture 2.4 (Tait’s conjecture IV) Amphicheiral (alternating?) knots have even crossing number.
Note that (for alternating knots, and with the remark in footnote 1) this is a consequence of conjecture II, for
mirroring a diagram interchanges positive and negative crossings, and so negates the writhe.
3. Reidemeister moves and invariants
A few decades after their genesis, Tait’s knot lists would be proved right. With the work of Alexander, Reidemeister
and others knot theory began to be put on a mathematical fundament. Reidemeister showed that three types of local
moves (i.e. moves altering only a fragment of the diagrams, as shown below) suffice to interrelate all diagrams of
the same knot.
←→ ←→ ←→ ←→ ←→ ←→
To distinguish two knots thus translates into the question how to prove that two diagrams (of these knots) are not
connected by a sequence of such moves. This is done with the help of an invariant, that is, a map
{ knot diagrams }→ “something” ,
whose value in “something” does not change (is invariant) when the argument (diagram) is changed by a Reide-
meister move. The question what “something” should be is justified. The answer is that it would suffice to be any
set of objects whose distinctness is easy to verify, yet which is large enough to allow the invariant to take many
different values. For us it will be the ring Z[t, t−1] of Laurent polynomials in one variable with integer coefficients.
Clearly one can compare coefficients easier than wondering about a sequence of Reidemeister moves, which may
be arbitrarily long and pass over arbitrarily complicated intermediate diagrams. (There is a fundamental, but out
of our focus here, issue how to, and that one can to some extent, control these sequences [HL].) An obvious other
desirable feature of an invariant is that we could easily evaluate it from a diagram.
Alexander’s merit was to construct precisely such an invariant. The Alexander polynomial [Al] remained (and still
remains) a main theme in knot theory for decades to come. Let us observe, though, that we actually already came
across one other knot invariant, the crossing number. It is an invariant directly by definition, since it was defined on
the whole Reidemeister move equivalence class of diagrams. However, this definition makes it difficult to evaluate
from a diagram – in contrast to Alexander’s polynomial, for which its creator, and later many others, gave several
simple procedures.
4. The Jones polynomial
60 years after Alexander, a new chapter of knot theory was opened by V. Jones, with the discovery of a successor
to Alexander’s polynomial. The developments the Jones polynomial V [J] has sparked in the 20 years since its
appearance are impossible even to be vaguely sketched in completeness, and go far beyond both the competence
and expository intention of the author here. Several more concepts, like links, braids, geometric, Vassiliev and
quantum invariants etc., are left out for simplicity and length reasons, for which the author likes to apologise at this
point. A good (though still partial, and now no longer very recent) account on these issues was given by Birman
[Bi].
Let us recall, though, one of the first achievements the Jones polynomial became famous with – the solution of Tait’s
conjectures. Conjectures I, II and IV, for alternating knots, were proved by Kauffman [Kf2], Murasugi [Mu, Mu2]
and Thistlethwaite [Th2]. We will need a few more words on Kauffman’s proof, since it uses a calculation procedure
for V called state model. This state model also gives a very elementary proof that the Jones polynomial is an
invariant. (Kauffman had previously developed a similar model for the Alexander polynomial, too [Kf3].)
Based on Kauffman’s state model, Lickorish and Thistlethwaite [LT] defined a (semi)adequate knot and diagram.
An advantage of this concept is that alternating diagrams/knots are adequate. Many details would better be skipped
here, but let us say that a diagram is adequate/semiadequate if it is +adequate and/or−adequate, and that taking the
mirror image of a diagram transforms the property for + into the one for −. A knot with some of these properties
4 6 Semiadequacy invariants and the non-triviality problem
is defined as one that has a diagram with the feature of the same name. Thistlethwaite extended the proof of the
Tait’s three conjectures to the class of adequate knots [Th], applying a 2-variable generalisation of V , the Kauffman
polynomial [Kf].
For a general (non-alternating) knot, in case of conjecture II, Tait’s intuition had been proved misleading. In
the 1970’s, K. Perko [Pe] observed that there are two 10 crossing knots in the tables [Ro, appendix], which are
equivalent, even though their 10 crossing diagrams have different writhe. This duplication had remained unnoticed
for quite a while, possibly due to the belief in Tait’s conjecture. Remedying this error (and a few other long-
remained ones) in the tables still causes some confusion in their use.
Tait’s conjecture III was settled a few years after the others by Menasco and Thistlethwaite [MT], mostly using
geometric techniques, though again with some (now subordinate) appearance of the Jones polynomial.
5. The crossing numbers of amphicheiral knots
Kauffman, Murasugi and Thistlethwaite’s proof of Tait’s conjecture IV shows that an alternating knot K of odd
crossing number and its mirror image !K always have distinct Jones polynomials. (Let us in contrast remark that K
and !K have the same Alexander polynomial for every knot K.) In the opposite direction, their work allows to easily
find a(n alternating) amphicheiral knot of every even crossing number at least 4. But their results could not decide
what crossing number non-alternating amphicheiral knots can have. The main aim of this note is to announce the
complete solution to Tait’s (last?) problem.
Theorem 5.1 For each odd natural number n≥ 15, there exists an amphicheiral knot of crossing number n.
Similarly to Perko’s knot, a particular instance disproving Tait’s conjecture IV for non-alternating knots was found
accidentally: Hoste and Thistlethwaite, in the course of routine knot tabulation, discovered an amphicheiral 15
crossing knot. (Their compilational work had previously shown that there are no amphicheiral knots of odd crossing
numbers up to 13.) Settling the other crossing numbers is a major problem, though, since exhaustive enumeration
is no longer a feasible approach – we face the above noticed difficulty that we do not know (generally) how to
determine the crossing number. A few other methods are known, but all they fail on such examples. Thus the
way to our result is rather far, and below we will conclude by just giving a brief outline of the proof. The details
will appear in a separate (long) paper. We also mention another application of our approach, which addresses the
non-triviality problem for the Jones polynomial.
6. Semiadequacy invariants and the non-triviality problem
The Alexander polynomial was, from its very beginning, connected to topological features of knots. The situation
is rather different for its successor. The problem to give a topological meaning to the Jones polynomial has bothered
many knot theorists ever since this invariant appeared. So far we still find ourselves in the embarrassing state where
we “can quickly fill pages with the coefficients and exponents of V for not-too-complicated knots without having
the slightest idea what they mean” ([Bi, end of §3]). Similarly unsolved, and intriguing, remains the problem,
formulated by Jones, if his polynomial detects the unknot. Again, “our lack of knowledge about this problem
is in striking contrast to the control mathematicians now have over the Alexander polynomial: understanding its
topological meaning, we also know precisely how to construct knots with Alexander polynomial 1” (ibid., rem. (iii)
after theorem 3, §8; see, though, also [EKT]).
In an attempt to gain more insight into the appearance of the Jones polynomial, the author, and (up to minor
interaction, independently) Dasbach and Lin [DL, DL2], initiated a detailed study of some coefficients of V in
semiadequate diagrams. Let us remark here that, while adequacy is only a slight extension of alternation, semiad-
equacy is a rather wide extension of adequacy. (For the experts: semiadequate knots contain completely positive,
Montesinos and 3-braid knots.) Semiadequacy is still a fairly general condition, yet it helps settle many technical
issues.
For semiadequate knots the first coefficient of the Jones polynomial is ±1, almost by definition [LT]. The outcome
of our work was that we gained an understanding of coefficients 2 and 3. Their invariance allows to derive 3
invariant quantities each from a +adequate, and similarly from a −adequate diagram, called below semiadequacy
References 5
invariants. Their merit is that they reflect directly certain features of the diagram, and so we have a precise idea
how a semiadequate diagram with given invariants must look like. The first of them allows to prove:
Theorem 6.1 Semiadequate knots have non-trivial Jones polynomial.
This implies (for experts) the result also for Montesinos and 3-braid knots, but it can be proved also for their
Whitehead doubles, some strongly n-trivial knots and k-almost positive knots with k ≤ 3.
Our three semiadequacy invariants become also, joined by a relative obtained from the Kauffman polynomial and
Thistlethwaite’s results [Th], the main tool for the proof of theorem 5.1. For given odd n ≥ 15, we start with
an amphicheiral knot K that has an n crossing diagram, which is semiadequate. Luckily, such examples can be
obtained by leaning on Hoste-Thistlethwaite’s knot. The work in [Th] shows then that the crossing number of K
is at least n− 1, and were it n− 1, a minimal crossing diagram D would be adequate. Then we have 4 invariants
for both +adequacy and −adequacy each available. A detailed study of how an n− 1 crossing diagram with such
invariants must look like is necessary to exclude most cases for D. Hereby, among the various generalisations
of Thistlethwaite’s knot, one must choose carefully the one whose invariants make the exclusion argument most
convenient (or better to say, feasible at all). Only a small fraction of possibilities for D remain, which are easy to
check, and rule out, by computer. This allows us to conclude that in fact D cannot exist.
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