L11a84

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{3 u v^3-9 u v^2+10 u v-5 u-5 v^3+10 v^2-9 v+3}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+8 q^{11/2}-12 q^{9/2}+16 q^{7/2}-17 q^{5/2}+17 q^{3/2}-15 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + 2 z 5 a -3 -2 a z 3 - z 3 a -1 + 4 z 3 a -3 -3 z 3 a -5 + a 3 z -2 a z -3 z a -1 + 4 z a -3 -3 z a -5 + z a -7 + a 3 z -1 -2 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -8 z 8 a -2 -11 z 8 a -4 -6 z 8 a -6 -3 z 8 -3 a z 7 + z 7 a -1 + 9 z 7 a -3 + z 7 a -5 -4 z 7 a -7 -2 a 2 z 6 + 19 z 6 a -2 + 33 z 6 a -4 + 14 z 6 a -6 - z 6 a -8 - z 6 - a 3 z 5 + 3 a z 5 + 3 z 5 a -3 + 17 z 5 a -5 + 10 z 5 a -7 + 3 a 2 z 4 -14 z 4 a -2 -28 z 4 a -4 -7 z 4 a -6 + 2 z 4 a -8 + 8 z 4 + 3 a 3 z 3 -2 z 3 a -1 -7 z 3 a -3 -14 z 3 a -5 -6 z 3 a -7 -4 z 2 a -2 + 7 z 2 a -4 + 2 z 2 a -6 - z 2 a -8 -8 z 2 -3 a 3 z + 3 z a -1 + z a -3 + 2 z a -5 + z a -7 - a 2 + 5 a -2 + 2 a -4 + 3 + a 3 z -1 -2 a -1 z -1 - a -3 z -1$ (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 L11a84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$