L11n442

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -2 v u 2 + v x u 2 - v w x u 2 + w x u 2 - x u 2 + u 2 -2 v 2 u + 4 v u + v 2 w u -2 v w u -2 v x u - v 2 w x u + 4 v w x u -2 w x u + x u - u + v 2 - v - v 2 w + v w + v 2 w x -2 v w x + w x$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+6 q^{11/2}-10 q^{9/2}+10 q^{7/2}-14 q^{5/2}+10 q^{3/2}-10 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{3}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2 z 5 a -3 -5 z 3 a -1 + 7 z 3 a -3 -3 z 3 a -5 + 3 a z -11 z a -1 + 13 z a -3 -6 z a -5 + z a -7 + 3 a z -1 -9 a -1 z -1 + 10 a -3 z -1 -5 a -5 z -1 + a -7 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -5 z 8 a -2 -8 z 8 a -4 -3 z 8 a -6 -7 z 7 a -1 -14 z 7 a -3 -10 z 7 a -5 -3 z 7 a -7 + 7 z 6 a -2 + 14 z 6 a -4 + 3 z 6 a -6 - z 6 a -8 -3 z 6 + 22 z 5 a -1 + 53 z 5 a -3 + 40 z 5 a -5 + 9 z 5 a -7 + 4 z 4 a -2 + 9 z 4 a -4 + 11 z 4 a -6 + 3 z 4 a -8 + 3 z 4 -6 a z 3 -42 z 3 a -1 -74 z 3 a -3 -47 z 3 a -5 -9 z 3 a -7 -24 z 2 a -2 -28 z 2 a -4 -14 z 2 a -6 -3 z 2 a -8 -7 z 2 + 10 a z + 35 z a -1 + 49 z a -3 + 30 z a -5 + 6 z a -7 + 21 a -2 + 18 a -4 + 6 a -6 + a -8 + 9-5 a z -1 -14 a -1 z -1 -18 a -3 z -1 -11 a -5 z -1 -2 a -7 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 L11n442. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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