L11a162

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,11,17,12 X_14,6,15,5 X_4,16,5,15 X_20,13,21,14 X_22,17,7,18 X_18,21,19,22 X_12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -11, 8, -6, 7, -5, 9, -10, 11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 4 v 2 u 3 -4 v u 3 + 2 u 3 -4 v 2 u 2 + 5 v u 2 -4 u 2 + 2 v 2 u -4 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{13}{q^{13/2}}+\frac{12}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -3 z 3 a 9 -2 z a 9 - a 9 z -1 + z 7 a 7 + 4 z 5 a 7 + 5 z 3 a 7 + 4 z a 7 + 2 a 7 z -1 + z 7 a 5 + 4 z 5 a 5 + 4 z 3 a 5 + z a 5 - z 5 a 3 -4 z 3 a 3 -4 z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$- z 4 a 14 + z 2 a 14 -3 z 5 a 13 + 3 z 3 a 13 -5 z 6 a 12 + 6 z 4 a 12 -2 z 2 a 12 -6 z 7 a 11 + 9 z 5 a 11 -6 z 3 a 11 + z a 11 -6 z 8 a 10 + 13 z 6 a 10 -15 z 4 a 10 + 8 z 2 a 10 -2 a 10 -4 z 9 a 9 + 8 z 7 a 9 -8 z 5 a 9 + 6 z 3 a 9 -3 z a 9 + a 9 z -1 - z 10 a 8 -6 z 8 a 8 + 27 z 6 a 8 -36 z 4 a 8 + 22 z 2 a 8 -5 a 8 -6 z 9 a 7 + 20 z 7 a 7 -23 z 5 a 7 + 16 z 3 a 7 -7 z a 7 + 2 a 7 z -1 - z 10 a 6 -2 z 8 a 6 + 17 z 6 a 6 -22 z 4 a 6 + 12 z 2 a 6 -3 a 6 -2 z 9 a 5 + 5 z 7 a 5 + 2 z 5 a 5 -7 z 3 a 5 + 2 z a 5 -2 z 8 a 4 + 8 z 6 a 4 -8 z 4 a 4 + z 2 a 4 + a 4 - z 7 a 3 + 5 z 5 a 3 -8 z 3 a 3 + 5 z a 3 - a 3 z -1$ (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 =

-5 is the signature of L11a162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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