L11n450

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + v w u 3 - w u 3 + v x u 3 - v w x u 3 + u 3 + 2 v u 2 -2 v w u 2 + 3 w u 2 -3 v x u 2 + 3 v w x u 2 - w x u 2 + x u 2 -3 u 2 - v u + v w u -3 w u + 3 v x u -3 v w x u + 2 w x u -2 x u + 3 u + w - v x + v w x - w x + x -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+7 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 8 a z 3 -6 z 3 a -1 + z 3 a -3 + a 5 z -7 a 3 z + 12 a z -8 z a -1 + 2 z a -3 + 2 a 5 z -1 -8 a 3 z -1 + 11 a z -1 -6 a -1 z -1 + a -3 z -1 + a 5 z -3 -3 a 3 z -3 + 3 a z -3 - a -1 z -3$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -9 a 2 z 8 -4 z 8 a -2 -13 z 8 -13 a 3 z 7 -20 a z 7 -10 z 7 a -1 -3 z 7 a -3 -6 a 4 z 6 + 9 a 2 z 6 + 5 z 6 a -2 - z 6 a -4 + 21 z 6 + 31 a 3 z 5 + 62 a z 5 + 39 z 5 a -1 + 8 z 5 a -3 + 4 a 4 z 4 + 9 a 2 z 4 + 7 z 4 a -2 + 3 z 4 a -4 + 9 z 4 -10 a 5 z 3 -44 a 3 z 3 -67 a z 3 -42 z 3 a -1 -9 z 3 a -3 -9 a 4 z 2 -24 a 2 z 2 -14 z 2 a -2 -3 z 2 a -4 -26 z 2 + 11 a 5 z + 34 a 3 z + 44 a z + 27 z a -1 + 6 z a -3 + 9 a 4 + 21 a 2 + 6 a -2 + a -4 + 18-5 a 5 z -1 -14 a 3 z -1 -18 a z -1 -11 a -1 z -1 -2 a -3 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L11n450. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -4$	${\mathbb Z}^{4}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$