L11a69

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,10,15,9 X_22,20,5,19 X_20,11,21,12 X_10,21,11,22 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -11 v u 3 + 11 u 3 + 11 v u 2 -11 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 -3 z 5 a 3 -8 z 3 a 3 -8 z a 3 -3 a 3 z -1 + z 7 a + 4 z 5 a + 8 z 3 a + 6 z a + 2 a z -1 - z 5 a -1 -2 z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -4 a 5 z 9 -9 a 3 z 9 -5 a z 9 -5 a 6 z 8 -16 a 4 z 8 -20 a 2 z 8 -9 z 8 -3 a 7 z 7 -2 a 5 z 7 -9 a z 7 -8 z 7 a -1 - a 8 z 6 + 9 a 6 z 6 + 41 a 4 z 6 + 47 a 2 z 6 -4 z 6 a -2 + 12 z 6 + 7 a 7 z 5 + 21 a 5 z 5 + 36 a 3 z 5 + 36 a z 5 + 13 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 -3 a 6 z 4 -32 a 4 z 4 -37 a 2 z 4 + 5 z 4 a -2 -6 z 4 -6 a 7 z 3 -22 a 5 z 3 -41 a 3 z 3 -35 a z 3 -9 z 3 a -1 + z 3 a -3 -3 a 8 z 2 -2 a 6 z 2 + 9 a 4 z 2 + 11 a 2 z 2 - z 2 a -2 + 2 z 2 + 3 a 7 z + 12 a 5 z + 18 a 3 z + 13 a z + 4 z a -1 + a 8 + 2 a 6 -2 a 2 - a 7 z -1 -3 a 5 z -1 -3 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 =

-1 is the signature of L11a69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$