L11n108

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,19,14,18 X_17,11,18,10 X_21,9,22,8 X_7,17,8,16 X_9,21,10,20 X_15,5,16,22 X_19,15,20,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 u 5 - v u 4 - u 4 + 3 v u 3 -2 u 3 -2 v u 2 + 3 u 2 - v u - u + 2 v$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+3 q^{11/2}-3 q^{9/2}+3 q^{7/2}-2 q^{5/2}+q^{3/2}-\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + a z 3 -5 z 3 a -1 - z 3 a -3 - z 3 a -5 + 3 a z -5 z a -1 - z a -3 + z a -7 + 2 a z -1 -2 a -1 z -1 - a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -2 z 8 a -2 -2 z 8 a -4 - z 8 a -6 - z 8 - a z 7 + 6 z 7 a -1 + 6 z 7 a -3 -3 z 7 a -5 -2 z 7 a -7 + 14 z 6 a -2 + 12 z 6 a -4 + 2 z 6 a -6 - z 6 a -8 + 5 z 6 + 6 a z 5 -7 z 5 a -1 -8 z 5 a -3 + 13 z 5 a -5 + 8 z 5 a -7 -24 z 4 a -2 -21 z 4 a -4 + 4 z 4 a -6 + 4 z 4 a -8 -3 z 4 -10 a z 3 -2 z 3 a -1 + 3 z 3 a -3 -12 z 3 a -5 -7 z 3 a -7 + 10 z 2 a -2 + 15 z 2 a -4 -3 z 2 a -6 -4 z 2 a -8 -4 z 2 + 7 a z + 4 z a -1 -3 z a -3 + 2 z a -5 + 2 z a -7 -3 a -4 + a -8 + 3-2 a z -1 -2 a -1 z -1 + a -3 z -1 + a -5 z -1$ (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 L11n108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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