Basic graph theory

The first non-alternating achiral link is the 4-component link 10-^***, or more precisely 10^***-1.-1.-1.-1::.-1, i.e., the basic polyhedron 10^*** turned by suitable crossing changes into the non-alternating link. From it the following infinite families are obtained:

the non-alternating 4-component links 10^***-(2m+1).-(2n+1).-(2p+1).-(2q+1).(2n+1).(2m+1).(2q+1).(2p+1).-(2r+1).(2r+1), (m,n,p,q,r ³ 1);

the non-alternating achiral 2-component links 10^***-(2m).-(2n).-(2p).-(2q).(2n).(2m).(2q).(2p).-(2r).(2r), (m,n,p,q,r ³ 1);

the non-alternating achiral 2-component links 10^***-(2m) -1.-(2n) -1.-(2p) -1.-(2q) -1.(2n) 1.(2m) 1.(2q) 1.(2p) 1.-(2r) -1.(2r) 1, (m,n,p,q,r ³ 1), etc..

In fact, an infinite collection of achiral KLs can be derived from the non-alternating achiral link 10-^*** by any tangle replacement of the form 10^***-t₁.-t₂.-t₃.-t₄.t₂.t₁.t₄.t₃.-t₅.t₅, where t_i are arbitrary tangles (i=1,2,...,5). A short proof of that property is: symmetrical replacements in an (anti)symmetrical structure result in the preservation of (anti)symmetry.

Because every (anti)symmetrical structure can be obtained from another (anti)symmetrical structure by a series of symmetrical replacements, the origins of achiral KLs are lower level achirals. For example, polyhedral achiral alternating KLs originate from achiral basic polyhedra and can be obtained by symmetrical tangle replacements.

Every palindromic rational knot with an even number of crossings is achiral, and we conjectured that the same statement holds for rational links. Trying to generalize, we can conclude that every algebraic alternating KL of the symmetrical (palindromic) form p p is achiral, where p is any algebraic tangle, and we believe that all achiral alternating algebraic KLs are described in this way.

For achiral alternating polyhedral KLs, the origin will be achiral basic polyhedra. Their list for n £ 16 is:

n=6

6^*

No. of basic polyhedra: 1

n=8

8^*

No. of basic polyhedra: 1

n=10

10^*	10^**	10^***
No. of basic polyhedra: 3

n=12

121^*	122^*	127^*	128^*	1210^*	1211^*	1212^*
No. of basic polyhedra: 7

n=14
144^*	147^*	148^*	149^*	1411^*	1412^*	1413^*	1417^*
1419^*	1420^*	1428^*	1433^*	1444^*	1447^*	1449^*	1450^*
1453^*	1455^*	1456^*	1458^*	1460^*
No. of basic polyhedra: 21

n=16

162^*	167^*	1617^*	1621^*	1623^*	1625^*	1632^*	1639^*
1643^*	1646^*	1647^*	1648^*	1649^*	1651^*	1674^*	1680^*
1685^*	1686^*	1689^*	1692^*	1697^*	16110^*	16113^*	16128^*
16132^*	16133^*	16142^*	16150^*	16156^*	16160^*	16175^*	16204^*
16206^*	16223^*	16226^*	16227^*	16230^*	16234^*	16235^*	16239^*
16242^*	16256^*	16263^*	16270^*	16273^*	16280^*	16282^*	16284^*
16285^*	16286^*	16293^*	16327^*	16346^*	16347^*	16351^*	16360^*
16361^*	16367^*	16369^*	16374^*	16377^*	16380^*	16384^*	16388^*
16391^*	16393^*	16402^*	16412^*	16416^*	16419^*	16428^*	16429^*
16431^*	16437^*	16442^*	16451^*	16461^*	16494^*
No. of basic polyhedra: 78