L11n171

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_11,21,12,20 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_13,19,14,18 X_22,16,7,15 X_19,13,20,12 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -6 v u 3 + u 3 -3 v 2 u 2 + 9 v u 2 -3 u 2 + v 2 u -6 v u + 3 u + v -1$ (db)
Jones polynomial	$2 q^{11/2}-5 q^{9/2}+9 q^{7/2}-12 q^{5/2}+13 q^{3/2}-14 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -4 z 5 a -1 + z 5 a -3 + 2 a z 3 -5 z 3 a -1 + z 3 a -3 + a z - z a -1 -2 z a -3 + z a -5 + a z -1 -2 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -1 -2 z 9 a -3 -9 z 8 a -2 -3 z 8 a -4 -6 z 8 -7 a z 7 -6 z 7 a -1 - z 7 a -5 -4 a 2 z 6 + 20 z 6 a -2 + 6 z 6 a -4 + 10 z 6 - a 3 z 5 + 14 a z 5 + 18 z 5 a -1 -3 z 5 a -5 + 6 a 2 z 4 -22 z 4 a -2 -14 z 4 a -4 -3 z 4 a -6 -5 z 4 + a 3 z 3 -7 a z 3 -12 z 3 a -1 + z 3 a -3 + 5 z 3 a -5 + 12 z 2 a -2 + 15 z 2 a -4 + 5 z 2 a -6 + 2 z 2 + 2 a z -3 z a -3 - z a -5 -3 a -2 -5 a -4 -2 a -6 + 1- a z -1 + 2 a -3 z -1 + a -5 z -1$ (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 L11n171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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