L11a422

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - v w u 4 + w u 4 - u 4 + 2 v 2 u 3 -5 v u 3 - v 2 w u 3 + 5 v w u 3 -3 w u 3 + 2 u 3 -3 v 2 u 2 + 6 v u 2 + 2 v 2 w u 2 -6 v w u 2 + 3 w u 2 -2 u 2 + 3 v 2 u -5 v u -2 v 2 w u + 5 v w u -2 w u + u - v 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$1-4 q -1 + 10 q -2 -14 q -3 + 21 q -4 -22 q -5 + 23 q -6 -19 q -7 + 14 q -8 -8 q -9 + 3 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 -2 a 10 + 3 z 4 a 8 + 8 z 2 a 8 + 4 a 8 z -2 + 9 a 8 -2 z 6 a 6 -7 z 4 a 6 -13 z 2 a 6 -5 a 6 z -2 -14 a 6 - z 6 a 4 + 6 z 2 a 4 + 2 a 4 z -2 + 7 a 4 + z 4 a 2 + z 2 a 2$ (db)
Kauffman polynomial	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -4 z 4 a 12 + z 2 a 12 + 6 z 7 a 11 -9 z 5 a 11 + 7 z 3 a 11 -4 z a 11 + a 11 z -1 + 8 z 8 a 10 -13 z 6 a 10 + 14 z 4 a 10 -11 z 2 a 10 - a 10 z -2 + 5 a 10 + 6 z 9 a 9 -20 z 5 a 9 + 33 z 3 a 9 -21 z a 9 + 5 a 9 z -1 + 2 z 10 a 8 + 16 z 8 a 8 -51 z 6 a 8 + 65 z 4 a 8 -44 z 2 a 8 -4 a 8 z -2 + 18 a 8 + 13 z 9 a 7 -20 z 7 a 7 -7 z 5 a 7 + 34 z 3 a 7 -29 z a 7 + 9 a 7 z -1 + 2 z 10 a 6 + 16 z 8 a 6 -56 z 6 a 6 + 67 z 4 a 6 -47 z 2 a 6 -5 a 6 z -2 + 21 a 6 + 7 z 9 a 5 -10 z 7 a 5 -5 z 5 a 5 + 13 z 3 a 5 -13 z a 5 + 5 a 5 z -1 + 8 z 8 a 4 -20 z 6 a 4 + 18 z 4 a 4 -14 z 2 a 4 -2 a 4 z -2 + 9 a 4 + 4 z 7 a 3 -8 z 5 a 3 + 3 z 3 a 3 + z 6 a 2 -2 z 4 a 2 + z 2 a 2$ (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 =

-4 is the signature of L11a422. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$