L11n116

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_7,17,8,16 X_20,17,21,18 X_18,13,19,14 X_14,19,15,20 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_15,9,16,8
Gauss code	{1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, 5, -3, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 4 u 4 + 7 v u 3 -7 u 3 -7 v u 2 + 7 u 2 + 4 v u -2 u - v$ (db)
Jones polynomial	$-\frac{3}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 + a 9 z -1 + z 5 a 7 -3 z a 7 -3 a 7 z -1 + 2 z 5 a 5 + 5 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 -3 z 3 a 3 -5 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z 6 a 12 + 2 z 4 a 12 - z 2 a 12 -4 z 7 a 11 + 11 z 5 a 11 -8 z 3 a 11 -5 z 8 a 10 + 11 z 6 a 10 -4 z 4 a 10 - z 2 a 10 + a 10 -2 z 9 a 9 -7 z 7 a 9 + 28 z 5 a 9 -19 z 3 a 9 + 4 z a 9 - a 9 z -1 -11 z 8 a 8 + 22 z 6 a 8 -7 z 4 a 8 -2 z 2 a 8 + 3 a 8 -2 z 9 a 7 -10 z 7 a 7 + 31 z 5 a 7 -27 z 3 a 7 + 12 z a 7 -3 a 7 z -1 -6 z 8 a 6 + 7 z 6 a 6 - z 4 a 6 -5 z 2 a 6 + 3 a 6 -7 z 7 a 5 + 14 z 5 a 5 -22 z 3 a 5 + 15 z a 5 -4 a 5 z -1 -3 z 6 a 4 -3 z 2 a 4 + 2 a 4 -6 z 3 a 3 + 7 z a 3 -2 a 3 z -1$ (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 =

-3 is the signature of L11n116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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