L11n162

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_17,13,18,12 X_14,5,15,6 X_4,13,5,14 X_11,19,12,18 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 - u 4 - v 2 u 3 + 2 v u 3 + v 2 u 2 -3 v u 2 + u 2 + 2 v u - u - v 2 - v$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+3 q^{5/2}-4 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 - z 5 a -4 z 3 a -5 z a -2 a z -1 - z 5 a -1 -3 z 3 a -1 - z a -1 + a -1 z -1 + z 3 a -3 + 2 z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -2 z 8 a -2 -3 z 8 - a 5 z 7 + 6 a z 7 + 3 z 7 a -1 -2 z 7 a -3 + 7 a 2 z 6 + 8 z 6 a -2 - z 6 a -4 + 16 z 6 + 7 a 5 z 5 + 2 a 3 z 5 -13 a z 5 + 8 z 5 a -3 + 2 a 4 z 4 -15 a 2 z 4 -8 z 4 a -2 + 4 z 4 a -4 -29 z 4 -14 a 5 z 3 -6 a 3 z 3 + 13 a z 3 -2 z 3 a -1 -7 z 3 a -3 -4 a 4 z 2 + 10 a 2 z 2 + 5 z 2 a -2 -3 z 2 a -4 + 22 z 2 + 9 a 5 z + 3 a 3 z -8 a z + 2 z a -3 + a 4 -3 a 2 -2 a -2 -5- a 5 z -1 + 2 a z -1 + a -1 z -1$ (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 L11n162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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