L11a177

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 3 v 2 u 5 -4 v u 5 + u 5 -5 v 2 u 4 + 9 v u 4 -3 u 4 + 5 v 2 u 3 -11 v u 3 + 5 u 3 -3 v 2 u 2 + 9 v u 2 -5 u 2 + v 2 u -4 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+20 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -6 a z 7 + z 7 a -1 + 4 a 3 z 5 -13 a z 5 + 4 z 5 a -1 + 5 a 3 z 3 -11 a z 3 + 5 z 3 a -1 + a 3 z - a z + z a -1 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-3 a 2 z 10 -3 z 10 -8 a 3 z 9 -15 a z 9 -7 z 9 a -1 -10 a 4 z 8 -11 a 2 z 8 -7 z 8 a -2 -8 z 8 -8 a 5 z 7 + 6 a 3 z 7 + 28 a z 7 + 10 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 13 a 4 z 6 + 26 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 24 z 6 - a 7 z 5 + 11 a 5 z 5 + a 3 z 5 -20 a z 5 + 9 z 5 a -3 + 5 a 6 z 4 -5 a 4 z 4 -14 a 2 z 4 -6 z 4 a -2 + 2 z 4 a -4 -12 z 4 + a 7 z 3 -5 a 5 z 3 + 2 a 3 z 3 + 12 a z 3 -2 z 3 a -1 -6 z 3 a -3 -2 a 6 z 2 - a 4 z 2 - z 2 a -2 - z 2 a -4 - z 2 -2 a 3 z - a z + 2 z a -1 + z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L11a177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$