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P.G. Tait conjectured that every achiral KL must have an even
number of crossings, so neither P.G. Tait nor M.G. Haseman
considered the possibility of the existence of achiral knots with
an odd crossing number. The first oriented achiral non-alternating link
8*.-2 0.2 0.-2 0 with n=11 crossings was discovered
in 1998 (Liang, Mislow and Flapan, 1998). The achiral
non-alternating knot 10**2 0..2 0.-2.-1.2 0..2 0.-2.-1
with n=15 crossings was found by M. Thistlethwaite, who also
recognized several duplications in Haseman's tables. However,
Tait's Conjecture about achiral KLs holds for alternating KLs:
there is no alternating achiral KL with an odd number of
crossings.
The non-alternating achiral oriented link
with n=11 crossings has a
non-minimal antisymmetrical representation
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123*-2 0.-1.-1.2 0:.-2 0.-1.-1.2 0 |
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with n=16 crossings, that shows its achirality.
The non-alternating achiral knot
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10**2 0.2..-2 0..2 0.-1.-1.-1.-2 0 |
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with
n=15 crossings has a non-minimal antisymmetrical
representation
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10**-2 0.-1.-2 0.2:-2 0.-1.-2 0.2 |
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with
n=16 crossings. That property can be extended to an
infinite family of achiral knots with an odd number of crossings.
From the antisymmetrical representation
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10**-2 0.-1.-2 0.2:-2 0.-1.-2 0.2 |
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of the knot
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10**2 0:2 0.-2.-1.2 0:2 0.-2.-1 |
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we derive the
three-parameter family
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10**(-2p) 0.-1.(-2q) 0.(2r):(-2p) 0.- 1.(-2q) 0.(2r) |
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of
achiral knots with n=2p+2q+2r+9 crossings. For example, the
non-minimal chiral antisymmetrical representation
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10**-4 0.-1.-6 0.8:-4 0.-1.-6 0.8 |
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with n=40 crossings
can be reduced to the achiral knot with n=39 crossings given by
the Dowker code
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{{39}, { 8, -32, -34 , 64, -62, -60, -66,42, 40, 38, -68, -70 , 44, 46, 48, 50,-4, -6, 24, 18, 16, 14 2, 6 , 28 , 30 , 36, -72, -74, -76, -12, -10, -78}}. |
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