L11n438

(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_7,15,8,14 X_13,5,14,8 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 3 + w u 3 + x u 3 - u 3 - v u 2 + 3 v w u 2 -2 w u 2 + 2 v x u 2 -2 v w x u 2 + w x u 2 -3 x u 2 + 2 u 2 + v u -3 v w u + 2 w u -2 v x u + 2 v w x u - w x u + 3 x u -2 u + v w + v x - v w x - x$ (db)
Jones polynomial	$q 21 / 2 -2 q 19 / 2 + 6 q 17 / 2 -10 q 15 / 2 + 12 q 13 / 2 -15 q 11 / 2 + 11 q 9 / 2 -13 q 7 / 2 + 6 q 5 / 2 -4 q 3 / 2$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-3 z 5 a -5 + 4 z 3 a -3 -12 z 3 a -5 + 6 z 3 a -7 + 9 z a -3 -22 z a -5 + 17 z a -7 -4 z a -9 + 7 a -3 z -1 -20 a -5 z -1 + 20 a -7 z -1 -8 a -9 z -1 + a -11 z -1 + 2 a -3 z -3 -7 a -5 z -3 + 9 a -7 z -3 -5 a -9 z -3 + a -11 z -3$ (db)
Kauffman polynomial	$- z 9 a -7 - z 9 a -9 -5 z 8 a -6 -7 z 8 a -8 -2 z 8 a -10 -10 z 7 a -5 -16 z 7 a -7 -8 z 7 a -9 -2 z 7 a -11 -6 z 6 a -4 -5 z 6 a -6 -2 z 6 a -10 - z 6 a -12 + 26 z 5 a -5 + 46 z 5 a -7 + 25 z 5 a -9 + 5 z 5 a -11 + 12 z 4 a -4 + 41 z 4 a -6 + 46 z 4 a -8 + 21 z 4 a -10 + 4 z 4 a -12 -10 z 3 a -3 -42 z 3 a -5 -50 z 3 a -7 -22 z 3 a -9 -4 z 3 a -11 -26 z 2 a -4 -73 z 2 a -6 -75 z 2 a -8 -34 z 2 a -10 -6 z 2 a -12 + 16 z a -3 + 39 z a -5 + 37 z a -7 + 16 z a -9 + 2 z a -11 + 23 a -4 + 60 a -6 + 58 a -8 + 24 a -10 + 4 a -12 -9 a -3 z -1 -23 a -5 z -1 -24 a -7 z -1 -12 a -9 z -1 -2 a -11 z -1 -7 a -4 z -2 -19 a -6 z -2 -18 a -8 z -2 -7 a -10 z -2 - a -12 z -2 + 2 a -3 z -3 + 7 a -5 z -3 + 9 a -7 z -3 + 5 a -9 z -3 + a -11 z -3$ (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 =

3 is the signature of L11n438. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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