L11a172

(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_18,11,19,12 X_14,6,15,5 X_20,13,21,14 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, -7, 10, -9, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 2 v 2 u 5 -3 v u 5 + u 5 -3 v 2 u 4 + 5 v u 4 -2 u 4 + 3 v 2 u 3 -5 v u 3 + 3 u 3 -2 v 2 u 2 + 5 v u 2 -3 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{5/2}+3 q^{3/2}-6 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{15}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 3 z 9 - a 5 z 7 + 7 a 3 z 7 - a z 7 -5 a 5 z 5 + 18 a 3 z 5 -5 a z 5 -8 a 5 z 3 + 20 a 3 z 3 -8 a z 3 -4 a 5 z + 8 a 3 z -5 a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 3 z 3 a 9 -5 z 6 a 8 + 5 z 4 a 8 - z 2 a 8 -7 z 7 a 7 + 12 z 5 a 7 -10 z 3 a 7 + 2 z a 7 -7 z 8 a 6 + 14 z 6 a 6 -12 z 4 a 6 + 2 z 2 a 6 -5 z 9 a 5 + 9 z 7 a 5 -5 z 5 a 5 + 2 z 3 a 5 -2 z a 5 -2 z 10 a 4 -2 z 8 a 4 + 16 z 6 a 4 -16 z 4 a 4 + 6 z 2 a 4 -9 z 9 a 3 + 33 z 7 a 3 -46 z 5 a 3 + 36 z 3 a 3 -11 z a 3 + a 3 z -1 -2 z 10 a 2 + 2 z 8 a 2 + 9 z 6 a 2 -11 z 4 a 2 + 5 z 2 a 2 - a 2 -4 z 9 a + 16 z 7 a -22 z 5 a + 17 z 3 a -7 z a + a z -1 -3 z 8 + 12 z 6 -13 z 4 + 3 z 2 - z 7 a -1 + 4 z 5 a -1 -4 z 3 a -1$ (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 L11a172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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