Knot polynomials

1 Princeton Companion to Mathematics Proof derived from distinguishing such groups Although rooted in algebraic topology, the Alexander polynomial has long been known to satisfy a skein By W B R Lickorish relation (see below) The HOMFLY polynomial of 1984 generalizes the Alexander polynomial and can Knots and Links be based on the simple combinatorics of skein theA knot is a curve in three-dimensional space that is ory alone closed (that is, it stops where it began) and never 1.1 The HOMFLY Polynomial meets itself along its way A link is several such curves, all disjoint from one another, which are Suppose that links are oriented so that directions, called the link’s components Some simple exam- indicated by arrows, are given to all components To each oriented link L is assigned its HOMFLY ples of knots and links are the following: polynomial P (L), a polynomial with integer coefficients in two variables v and z (allowing both positive and negative powers of v and z) The polynomials are such that unknot trefoil figure eight P (unknot) = (1.1) Knot Polynomials and there is a linear skein relation v −1 P (L+ ) − vP (L− ) = zP (L0 ) unlink Hopf link Whitehead link (1.2) This means that whenever three links have identiTwo knots are equivalent or “the same” if one cal diagrams except near one crossing, where they can be moved continuously, never breaking the are as follows “string,” to become the other “Isotopy” is the L− L0 , L+ technical term for such movement For example, the following knots are the same: then this equation holds This turns out to be good notation, although one could in principle use x and y in place of v −1 and −v Although Alexander’s polynomial satisfied a particular instance of (1.2), it took almost The first problem in knot theory is how to decide 60 years and the discovery of the Jones polynoif two knots are the same Two knots may appear mial for it to be realized that this general linear to be very different but how does one prove that relation can be used Note that there are two posthey are distinct? In classical geometry two tri- sible types of crossing in a diagram of an oriented angles are the same (or congruent) if one can be link A crossing is positive if, when approaching the moved rigidly on to the other Numbers, measur- crossing along the under-passing arc in the direcing side-lengths and angles, are assigned to each tion of the arrow, the other directed arc is seen to triangle to help determine if this might be Simi- cross over from left to right If the over-passing arc larly, mathematical entities, called invariants, can crosses from right to left, the crossing is negative be associated with knots and links in such a way When interpreting the skein relation at a crossing that, if two links have different invariants, then of a link L, it is vital that L be regarded as L+ if they cannot be the same link Many invariants the crossing is positive and as L− if it is negative relate to the geometry or topology of a link’s comThe theorem that underpins this theory, which plement in three-dimensional space The funda- is not at all obvious, is that it is possible to assign mental group of this complement is an excellent such polynomials to oriented links in a coherent invariant, but algebraic techniques are then needed fashion, uniquely, independent of any choice of a to distinguish the groups The polynomial of J W link’s diagram A proof of this is given in Lickorish Alexander (published in 1926) is a link invariant (1997) 2 Princeton Companion to Mathematics Proof every occurrence of v must be replaced by one −1 In a diagram of a knot it is always possible to of −v Thus the trefoil and its reflection, change some of the crossings, from over to under, to achieve a diagram of the unknot Links can be , undone similarly Using this, the polynomial of any link can be calculated from the above equations, though the length of the calculation is exponential have polynomials in the number of crossings The following is a calcu−4 −2 −2 and − v + 2v + z v lation of P (trefoil) Firstly, consider the following −v + 2v + z v 1.2 HOMFLY Calculations instance of the skein equation: v −1 P( ) − vP ( ) = zP ( ) As these polynomials are not the same, the trefoil and its reflection are different knots Substituting the polynomial for the polynomials Other Polynomial Invariants of the two unknots, this shows that the HOMFLY polynomial of the two-component unlink is The HOMFLY polynomial was inspired by the disz −1 (v −1 − v) A second usage of the skein equa- covery in 1984 of the polynomial of V F R Jones tion is For an oriented link L, the Jones polynomial V (L) has just one variable t (together with t−1 ) It is v −1 P − vP = zP obtained from P (L) by substituting v = t and z = t1/2 − t−1/2 , where t1/2 is just a formal square Substituting the previous answer for the unlink root of t The Alexander polynomial is obtained shows that the HOMFLY polynomial of the Hopf by the substitution v = 1, z = t−1/2 − t1/2 This link is z −1 (v −3 − v −1 ) − zv −1 Finally, consider latter polynomial is well understood in terms of the following instance of the skein equation: topology, by way of the fundamental group, covering spaces, and homology theory, and can be calcu− vP = zP v −1 P lated by various methods involving determinants It was J H Conway who, in discussing in 1969 Substitution of the polynomial for the Hopf link his normalized version of the Alexander polynoalready calculated and, of course, the value for mial (obtained by substituting v = and z = z), the unknot shows that first developed the theory of skein relations There is one more polynomial (due to L H −4 −2 −2 P (trefoil) = −v + 2v + z v Kauffman) based on a linear skein relation The relation involves four links with unoriented diaA similar calculation shows that grams differing as follows: P (figure eight) = v − + v −2 − z The trefoil and the figure eight thus have different polynomials; this proves they are different knots Experimentally, if a trefoil is actually made from a necklace (using the clasp to join the ends together) it is indeed found to be impossible to move it to the configuration of a figure eight knot Note that the polynomial of a knot is not dependent on the choice of its orientation (but this is not so for links) Reflecting a knot in a mirror is equivalent to changing every crossing in a diagram of the knot from an over-crossing to an under-crossing and vice versa (consider the plane of the diagram to be the mirror) The polynomial of the reflection is always the same as that of the original knot except that There are examples of pairs of knots that the Kauffman polynomial but not the HOMFLY polynomial can distinguish apart and vice versa; some pairs are not distinguished by any of these polynomials 2.1 Application to Alternating Knots For the Jones polynomial there is a particularly simple formulation, by means of “Kauffman’s bracket polynomial”, that leads to an easy proof that the Jones (but not the HOMFLY) polynomial is coherently defined This approach has been Princeton Companion to Mathematics Proof used to give the first rigorous confirmation of P G Tait’s (1898) highly believable proposal that a reduced alternating diagram of a knot has the minimal number of crossings for any diagram of that knot Here “alternating” means that in going along the knot the crossings go: over, under, over, under, over, Not every knot has such a diagram “Reduced” means that there are, adjacent to each crossing, four distinct regions of the diagram’s planar complement Thus, for example, any nontrivial reduced alternating diagram is not a diagram of the unknot Also, the figure eight knot certainly has no diagram with only three crossings 2.2 Physics Unlike that of Alexander, the HOMFLY polynomial has no known interpretation in terms of classical algebraic topology It can, however, be reformulated as a collection of state sums, summing over certain labelings of a knot diagram This recalls ideas from statistical mechanics; an elementary account is given in Kauffman (1991) An amplification of the whole HOMFLY polynomial theory leads into a version of conformal field theory called topological quantum field theory Further Reading Kauffman, L H 1991 Knots and Physics Singapore: World Scientific Lickorish, W B R 1997 An Introduction to Knot Theory Graduate Texts in Mathematics, Vol 175 New York: Springer Tait, P G 1898 On knots In Scientific Papers, Volume I, pp 273–347 Cambridge University Press ... ) As these polynomials are not the same, the trefoil and its reflection are different knots Substituting the polynomial for the polynomials Other Polynomial Invariants of the two unknots, this... configuration of a figure eight knot Note that the polynomial of a knot is not dependent on the choice of its orientation (but this is not so for links) Reflecting a knot in a mirror is equivalent... that a reduced alternating diagram of a knot has the minimal number of crossings for any diagram of that knot Here “alternating” means that in going along the knot the crossings go: over, under,