L11n118

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_9,16,10,17 X_11,21,12,20 X_21,9,22,8 X_7,19,8,18 X_19,13,20,12 X_15,10,16,11 X_17,5,18,22 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, -3, 8, -4, 7, -11, 2, -8, 3, -9, 6, -7, 4, -5, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 7 v u 2 -7 u 2 -7 v u + 7 u + 2 v -2$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+7 q^{11/2}-10 q^{9/2}+11 q^{7/2}-13 q^{5/2}+11 q^{3/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{2}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2 z 5 a -3 -4 z 3 a -1 + 6 z 3 a -3 -3 z 3 a -5 + 2 a z -6 z a -1 + 8 z a -3 -5 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 3 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -3 -2 z 9 a -5 -6 z 8 a -2 -10 z 8 a -4 -4 z 8 a -6 -5 z 7 a -1 -6 z 7 a -3 -4 z 7 a -5 -3 z 7 a -7 + 17 z 6 a -2 + 27 z 6 a -4 + 8 z 6 a -6 - z 6 a -8 - z 6 + 13 z 5 a -1 + 30 z 5 a -3 + 25 z 5 a -5 + 8 z 5 a -7 -24 z 4 a -2 -22 z 4 a -4 + z 4 a -6 + 3 z 4 a -8 -4 z 4 -3 a z 3 -23 z 3 a -1 -39 z 3 a -3 -25 z 3 a -5 -6 z 3 a -7 + 11 z 2 a -2 + 7 z 2 a -4 -3 z 2 a -6 -3 z 2 a -8 + 4 z 2 + 4 a z + 13 z a -1 + 18 z a -3 + 12 z a -5 + 3 z a -7 -2 a -2 + 2 a -6 + a -8 - a z -1 -2 a -1 z -1 -3 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11n118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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