L11n66

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,14,8,15 X_15,22,16,5 X_9,17,10,16 X_21,9,22,8 X_17,21,18,20 X_13,18,14,19 X_19,12,20,13 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 5 v u 4 - u 4 -8 v u 3 + 4 u 3 + 4 v u 2 -8 u 2 - v u + 5 u -1$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{2}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -1 - z 3 a 7 -4 z a 7 -3 a 7 z -1 + 2 z 5 a 5 + 7 z 3 a 5 + 8 z a 5 + 4 a 5 z -1 - z 7 a 3 -4 z 5 a 3 -6 z 3 a 3 -6 z a 3 -2 a 3 z -1 + z 5 a + 2 z 3 a$ (db)
Kauffman polynomial	$-3 z 3 a 9 + 4 z a 9 - a 9 z -1 - z 6 a 8 -4 z 4 a 8 + 4 z 2 a 8 - a 8 -5 z 7 a 7 + 13 z 5 a 7 -25 z 3 a 7 + 16 z a 7 -3 a 7 z -1 -6 z 8 a 6 + 16 z 6 a 6 -23 z 4 a 6 + 12 z 2 a 6 -3 a 6 -2 z 9 a 5 -8 z 7 a 5 + 37 z 5 a 5 -49 z 3 a 5 + 25 z a 5 -4 a 5 z -1 -11 z 8 a 4 + 29 z 6 a 4 -23 z 4 a 4 + 9 z 2 a 4 -2 a 4 -2 z 9 a 3 -7 z 7 a 3 + 35 z 5 a 3 -35 z 3 a 3 + 15 z a 3 -2 a 3 z -1 -5 z 8 a 2 + 11 z 6 a 2 -2 z 4 a 2 - a 2 -4 z 7 a + 11 z 5 a -8 z 3 a + 2 z a - z 6 + 2 z 4 - z 2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-3 is the signature of L11n66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -6$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$