L11n445

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_19,17,20,22 X_21,9,22,16 X_15,21,16,20 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-9, 5, -6, 8, -7, 6}, {-11, 2, -5, 9, -4, 3, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$x u 3 - u 3 - v u 2 + 2 v w u 2 - w u 2 + v x u 2 -2 v w x u 2 + w x u 2 -3 x u 2 + 3 u 2 + v u -3 v w u + w u - v x u + 3 v w x u - w x u + 2 x u -2 u + v w - v w x$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+6 q^{11/2}-10 q^{9/2}+9 q^{7/2}-12 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{2}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2 z 5 a -3 -4 z 3 a -1 + 7 z 3 a -3 -3 z 3 a -5 + 2 a z -9 z a -1 + 12 z a -3 -6 z a -5 + z a -7 + 3 a z -1 -9 a -1 z -1 + 10 a -3 z -1 -5 a -5 z -1 + a -7 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -4 z 8 a -2 -7 z 8 a -4 -3 z 8 a -6 -4 z 7 a -1 -10 z 7 a -3 -9 z 7 a -5 -3 z 7 a -7 + 8 z 6 a -2 + 13 z 6 a -4 + 3 z 6 a -6 - z 6 a -8 - z 6 + 14 z 5 a -1 + 43 z 5 a -3 + 38 z 5 a -5 + 9 z 5 a -7 + 9 z 4 a -4 + 12 z 4 a -6 + 3 z 4 a -8 -3 a z 3 -33 z 3 a -1 -66 z 3 a -3 -45 z 3 a -5 -9 z 3 a -7 -24 z 2 a -2 -30 z 2 a -4 -14 z 2 a -6 -3 z 2 a -8 -5 z 2 + 7 a z + 30 z a -1 + 48 z a -3 + 31 z a -5 + 6 z a -7 + 21 a -2 + 18 a -4 + 6 a -6 + a -8 + 9-5 a z -1 -14 a -1 z -1 -18 a -3 z -1 -11 a -5 z -1 -2 a -7 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 L11n445. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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