L11n182

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_14,5,15,6 X_16,8,17,7 X_3,10,4,11 X_22,14,7,13 X_18,12,19,11 X_12,18,13,17 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 4 - v 2 u 3 + 5 v u 3 -3 u 3 + 3 v 2 u 2 -7 v u 2 + 3 u 2 -3 v 2 u + 5 v u - u -2 v$ (db)
Jones polynomial	$2 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 + a 5 z -1 - z 5 a 3 - z 3 a 3 -2 z 5 a -6 z 3 a -7 z a -2 a z -1 + 2 z 3 a -1 + 3 z a -1 + a -1 z -1$ (db)
Kauffman polynomial	$-2 a 3 z 9 -2 a z 9 -5 a 4 z 8 -8 a 2 z 8 -3 z 8 -6 a 5 z 7 -2 a 3 z 7 + 3 a z 7 - z 7 a -1 -3 a 6 z 6 + 10 a 4 z 6 + 21 a 2 z 6 + 8 z 6 - a 7 z 5 + 16 a 5 z 5 + 10 a 3 z 5 -10 a z 5 -3 z 5 a -1 + 5 a 6 z 4 -6 a 4 z 4 -27 a 2 z 4 -3 z 4 a -2 -19 z 4 + 2 a 7 z 3 -16 a 5 z 3 -10 a 3 z 3 + 15 a z 3 + 7 z 3 a -1 - a 4 z 2 + 14 a 2 z 2 + 5 z 2 a -2 + 20 z 2 + 8 a 5 z + 2 a 3 z -9 a z -3 z a -1 + a 4 -3 a 2 -2 a -2 -5- a 5 z -1 + 2 a z -1 + a -1 z -1$ (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 L11n182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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