L11n177

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_5,15,6,14 X_18,12,19,11 X_10,4,11,3 X_7,13,8,12 X_13,17,14,16 X_17,7,18,22 X_15,1,16,6 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, -3, 9}, {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 2 v 2 u 3 -5 v u 3 + 3 u 3 -3 v 2 u 2 + 5 v u 2 -3 u 2 + 3 v 2 u -5 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-2 q^{15/2}+6 q^{13/2}-9 q^{11/2}+12 q^{9/2}-14 q^{7/2}+12 q^{5/2}-11 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 - z 5 a -1 + 4 z 5 a -3 - z 5 a -5 -2 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + z a -3 + z a -5 - z a -7 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -3 -2 z 9 a -5 -5 z 8 a -2 -10 z 8 a -4 -5 z 8 a -6 -4 z 7 a -1 -6 z 7 a -3 -6 z 7 a -5 -4 z 7 a -7 + 11 z 6 a -2 + 21 z 6 a -4 + 8 z 6 a -6 - z 6 a -8 - z 6 + 11 z 5 a -1 + 27 z 5 a -3 + 19 z 5 a -5 + 3 z 5 a -7 -2 z 4 a -2 -8 z 4 a -4 -10 z 4 a -6 -6 z 4 a -8 + 2 z 4 -8 z 3 a -1 -19 z 3 a -3 -12 z 3 a -5 -4 z 3 a -7 -3 z 3 a -9 -2 z 2 a -2 + 6 z 2 a -6 + 5 z 2 a -8 - z 2 + z a -1 + 2 z a -3 + z a -5 + z a -7 + z a -9 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11n177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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