L11a412

(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_14,10,15,9 X_20,16,21,15 X_10,19,5,20 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 7, -9}, {11, -2, 3, -7, 8, -5, 4, -3, 9, -8, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - v w u 4 + w u 4 - u 4 + 3 v 2 u 3 -6 v u 3 - v 2 w u 3 + 5 v w u 3 -4 w u 3 + 3 u 3 -5 v 2 u 2 + 9 v u 2 + 3 v 2 w u 2 -9 v w u 2 + 5 w u 2 -3 u 2 + 4 v 2 u -5 v u -3 v 2 w u + 6 v w u -3 w u + u - v 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$- q 2 + 5 q -12 + 20 q -1 -25 q -2 + 31 q -3 -28 q -4 + 25 q -5 -18 q -6 + 10 q -7 -4 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 -3 z 4 a 6 -5 z 2 a 6 + a 6 z -2 -3 a 6 + 2 z 6 a 4 + 5 z 4 a 4 + 6 z 2 a 4 -2 a 4 z -2 + a 4 + z 6 a 2 - z 2 a 2 + a 2 z -2 + a 2 - z 4$ (db)
Kauffman polynomial	$z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 4 z 7 a 9 -8 z 5 a 9 + 6 z 3 a 9 -2 z a 9 + 8 z 8 a 8 -16 z 6 a 8 + 13 z 4 a 8 -7 z 2 a 8 + 2 a 8 + 8 z 9 a 7 -6 z 7 a 7 -14 z 5 a 7 + 18 z 3 a 7 -7 z a 7 + 3 z 10 a 6 + 21 z 8 a 6 -64 z 6 a 6 + 59 z 4 a 6 -24 z 2 a 6 + a 6 z -2 + 4 a 6 + 20 z 9 a 5 -26 z 7 a 5 -13 z 5 a 5 + 25 z 3 a 5 -7 z a 5 -2 a 5 z -1 + 3 z 10 a 4 + 30 z 8 a 4 -79 z 6 a 4 + 63 z 4 a 4 -21 z 2 a 4 + 2 a 4 z -2 + 3 a 4 + 12 z 9 a 3 -4 z 7 a 3 -22 z 5 a 3 + 17 z 3 a 3 -3 z a 3 -2 a 3 z -1 + 17 z 8 a 2 -27 z 6 a 2 + 16 z 4 a 2 -5 z 2 a 2 + a 2 z -2 + 12 z 7 a -14 z 5 a + 4 z 3 a - z a + 5 z 6 -3 z 4 + z 5 a -1$ (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 =

-2 is the signature of L11a412. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -4$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -2$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{17}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$