L11a402

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + 2 v w u 4 - w u 4 -3 v 2 u 3 + 4 v u 3 + 2 v 2 w u 3 -5 v w u 3 + 3 w u 3 - u 3 + 4 v 2 u 2 -7 v u 2 -2 v 2 w u 2 + 7 v w u 2 -4 w u 2 + 2 u 2 -3 v 2 u + 5 v u + v 2 w u -4 v w u + 3 w u -2 u + v 2 -2 v + v w - w + 1$ (db)
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 16 q 2 -20 q + 25-23 q -1 + 21 q -2 -15 q -3 + 9 q -4 -4 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -7 a 2 z 4 -3 z 4 a -2 + 10 z 4 + 2 a 4 z 2 -8 a 2 z 2 -3 z 2 a -2 + 8 z 2 + a 4 -2 a 2 + 1 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a 2 z 10 + 2 z 10 + 6 a 3 z 9 + 14 a z 9 + 8 z 9 a -1 + 7 a 4 z 8 + 17 a 2 z 8 + 11 z 8 a -2 + 21 z 8 + 4 a 5 z 7 -3 a 3 z 7 -20 a z 7 -5 z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -15 a 4 z 6 -56 a 2 z 6 -20 z 6 a -2 + 4 z 6 a -4 -64 z 6 -9 a 5 z 5 -19 a 3 z 5 -9 a z 5 -11 z 5 a -1 -11 z 5 a -3 + z 5 a -5 -2 a 6 z 4 + 10 a 4 z 4 + 59 a 2 z 4 + 18 z 4 a -2 -5 z 4 a -4 + 70 z 4 + 6 a 5 z 3 + 21 a 3 z 3 + 25 a z 3 + 15 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + a 6 z 2 -6 a 4 z 2 -28 a 2 z 2 -9 z 2 a -2 + z 2 a -4 -31 z 2 -2 a 5 z -7 a 3 z -7 a z -3 z a -1 - z a -3 + 2 a 4 + 4 a 2 + 3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11a402. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$