L11a445

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_18,10,19,9 X_16,8,17,7 X_20,11,21,12 X_22,20,13,19 X_10,21,11,22 X_8,14,9,13 X_12,18,5,17 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -8, 3, -7, 5, -9}, {8, -2, 11, -4, 9, -3, 6, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + v 3 u 2 -5 v 2 u 2 + 4 v u 2 - v 3 w u 2 + 5 v 2 w u 2 -6 v w u 2 + 2 w u 2 - u 2 -2 v 3 u + 6 v 2 u -5 v u + v 3 w u -4 v 2 w u + 5 v w u - w u + u + v 3 -2 v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$- q 8 + 4 q 7 -9 q 6 + 14 q 5 -17 q 4 + 21 q 3 -19 q 2 + 17 q -11 + 7 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + z 4 a -2 + 2 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 - z 2 a -2 + 3 z 2 a -4 - z 2 a -6 -3 z 2 + a 2 -3 a -2 + 3 a -4 - a -6 -2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 3 z 9 a -1 + 8 z 9 a -3 + 5 z 9 a -5 + 11 z 8 a -2 + 16 z 8 a -4 + 9 z 8 a -6 + 4 z 8 + 3 a z 7 + 2 z 7 a -1 -6 z 7 a -3 + 3 z 7 a -5 + 8 z 7 a -7 + a 2 z 6 -31 z 6 a -2 -42 z 6 a -4 -15 z 6 a -6 + 4 z 6 a -8 -7 z 6 -8 a z 5 -17 z 5 a -1 -15 z 5 a -3 -20 z 5 a -5 -13 z 5 a -7 + z 5 a -9 -3 a 2 z 4 + 35 z 4 a -2 + 47 z 4 a -4 + 11 z 4 a -6 -5 z 4 a -8 + z 4 + 6 a z 3 + 17 z 3 a -1 + 24 z 3 a -3 + 20 z 3 a -5 + 6 z 3 a -7 - z 3 a -9 + 3 a 2 z 2 -26 z 2 a -2 -31 z 2 a -4 -7 z 2 a -6 + z 2 a -8 - a z -8 z a -1 -12 z a -3 -7 z a -5 -2 z a -7 - a 2 + 11 a -2 + 11 a -4 + 3 a -6 + 3 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11a445. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$