L11a174

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,8,17,7 X_22,16,7,15 X_6,21,1,22 X_20,11,21,12 X_14,6,15,5 X_18,13,19,14 X_12,19,13,20 X_4,18,5,17
Gauss code	{1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -9, 10, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 4 v 2 u 3 -7 v u 3 + 2 u 3 -4 v 2 u 2 + 11 v u 2 -4 u 2 + 2 v 2 u -7 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-7 q^{3/2}+11 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 4 a 3 z 5 + 4 a z 5 - z 5 a -1 -3 a 5 z 3 + 5 a 3 z 3 + 6 a z 3 -3 z 3 a -1 -2 a 5 z + 4 a z -3 z a -1 + a 5 z -1 -2 a 3 z -1 + 2 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -4 a 5 z 9 -9 a 3 z 9 -5 a z 9 -4 a 6 z 8 -3 a 2 z 8 -7 z 8 -3 a 7 z 7 + 9 a 5 z 7 + 26 a 3 z 7 + 8 a z 7 -6 z 7 a -1 - a 8 z 6 + 9 a 6 z 6 + 4 a 4 z 6 + 14 a 2 z 6 -3 z 6 a -2 + 17 z 6 + 9 a 7 z 5 -9 a 5 z 5 -35 a 3 z 5 -2 a z 5 + 14 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 -3 a 6 z 4 -2 a 4 z 4 -19 a 2 z 4 + 5 z 4 a -2 -18 z 4 -7 a 7 z 3 + 8 a 5 z 3 + 18 a 3 z 3 -11 a z 3 -12 z 3 a -1 + 2 z 3 a -3 -2 a 8 z 2 - a 6 z 2 + 2 a 4 z 2 + 7 a 2 z 2 + 6 z 2 + a 7 z + 2 a 3 z + 8 a z + 5 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11a174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$