DISCOVERING SYMMETRY OF KNOTS
Slavik JABLAN and RADMILA SAZDANOVIĆ
The research of amphicheirality started with works of J.B. Listing. He showed that the figure-eight knot, in honor of his accomplishment to knot theory called Listing knot is equal to its mirror image (this means, that figure-eight knot is amphicheiral), recognized that the left trefoil is different from the right trefoil knot (this means, that trefoil knot is chiral), and introduced a writhe of a knot, i.e., a signed sum of all crossings in a knot diagram.
The first results in the derivation of knots belong to P.G. Tait. The derivation of knots with 10 crossings took him six years to complete. In his works Tait also considered some of the fundamental problems in knot theory: chirality and unknotting number (or Gordian number, called by Tait "beknottedness"), and introduced the graph of a knot.
Thirty years after Tait's first results in enumeration of achiral knots with at most crossings, M.G. Hasseman in his dissertation partially extended knot tables, and described achiral knots with at most 12 crossings. Tait conjectured that every achiral knot or link must have an even number of crossings. Therefore neither Tait nor Haseman considered the possibility of the existence of achiral knots with an odd crossing number. The first oriented achiral link 8*.2 0.2 0. 2 0 with n=11 crossings was discovered in 1998. The achiral non-alternating knot 10**2 0:2 0. 2. 1.2 0:2 0. 2. 1 with n=15 crossings was recently found by M. Thistlethwaite, who also recognized several duplications in Haseman's tables. However, the Tait Conjecture about achiral knots or links holds for alternating KLs: there is no alternating achiral KL with an odd number of crossings.
In the case of rational knots all amphicheiral knots are derived from the same source: from the figure-eight knot 2 2. In general, a knot or link is achiral (or amphichiral) if its "left" and "right" forms are equivalent, meaning that one can be transformed to the other by an ambient isotopy. If a knot K could be represented by an antisymmetrical vertex-bicolored graph on a sphere, it is achiral In this case, for the oriented knot K there exist a symmetry transposing orientations of vertices, i.e., mutually exchanging vertices with the signs +1 and 1. Its antisymmetries (this means sign-changing symmetries) could be sense-reversing: rotational antireflection or anti-inversion, or sense-preserving: 2-antirotation. For the knot 2 2, the graph symmetry group is G = [2+, 4], and the knot symmetry group G' = [2+, 4+] is generated by the rotational reflection, with the axis defined by the midpoints of colored (i.e., double) edges of the tetrahedron. Considering the sign of the vertices, it is a rotational antireflection. Its effect is preserved in all rational knots with an even number of crossings that have a symmetrical Conway symbol. Hence, a rational knot is achiral iff its Conway symbol is symmetrical and has an even number of crossings. Achiral rational knots with at most 12 crossings are: 2 2 for n=4, 2 1 1 2 for n=6, 4 4, 3 1 1 3, and 2 2 2 2 for n=8, 1 1 4, 3 1 1 1 1 3, 2 3 3 2, 2 1 2 2 1 2, and 2 1 1 1 1 1 1 2 for n=10, and 6 6 , 5 1 1 5, 4 2 2 4, 3 3 3 3, 2 4 4 2, 3 2 1 1 2 3, 3 1 2 2 1 3, 2 2 2 2 2 2, 2 2 1 1 1 1 2 2 , 2 1 2 1 1 2 1 2, and 2 1 1 1 1 1 1 1 1 2 for n=12.
The LinKnot function RationalAmphiK calculates the number and Conway symbols of all rational achiral knots for a given number of crossings n. Calculating the number of achiral knots for n=2k (k=1, 2, 3, ) we obtain the Jacobsthal sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, ...defined by the recursion
a=0, a=1, 2a[k]+a[k+1]=a[k+2]
and given by the general formula
a[n]=(2n (1) n)/ 3
In recognizing other achiral knots we can use their antisymmetrical rigid representations in R3, or on the sphere S3 Every KL that has an antisymmetrical presentation in R3, has it on S3 as well, but not necessarily vice versa. For example, figure-eight knot has both presentations antisymmetrical: its non-minimal diagram is invariant with regard to a rotational antireflection of order 4 (i.e., rotational reflection followed by vertex sign change), and its diagram coming from S3 is centro-antisymmetrical. Achiral knot 817 has more remarkable property: it has a centro-antisymmetrical presentation coming from S3, but has no antisymmetrical presentation in R3. Therefore, it is a topological rubber glove in R3 (Flapan, 2000).
By using the function fDiffProjectionsAltKL we are able to calculate some invariants (or properties) of alternating KLs for which we need all non-isomorphic projections and to extend that properties from individual projections to the corresponding KLs. For example, if we find at least one achiral projection of a KL, we know that the KL in question is achiral. This mainly holds for the functions dealing with symmetry of KLs. In the same way as with graph automorphisms, we can deal with projections of KLs as with weighted graphs, where to every vertex is assigned its weight a sign of the vertex, and instead of classical automorphisms we consider automorphisms preserving signs. As before, those automorphisms make a group: the automorphism group of a KL.
The LinKnot function AmphiProjAltKL tests chirality of a given projection of an alternating KL given by its Conway symbol, Dowker code, or P-data, and the function AmphiAltKL tests the chirality of an alternating knot or link given by its Conway symbol. For example, the projection (((1,2,1,1),1,1,1),1,1) of the achiral knot 2 3 3 2 is not achiral, but about achirality of that knot we can conclude from the projection 2 3 3 2