L11n173

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_12,7,13,8 X_15,7,16,22 X_14,6,15,5 X_6,14,1,13 X_21,17,22,16 X_18,10,19,9 X_20,11,21,12 X_4,18,5,17 X_2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, -4, 7, 10, -8, 11, -9, -7, 4}

A Braid Representative

A Morse Link Presentation

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

1 is the signature of L11n173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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