L11a416

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -4 v u 2 - v 2 w u 2 + 3 v w u 2 -2 w u 2 + 2 u 2 -4 v 2 u + 8 v u + 3 v 2 w u -8 v w u + 4 w u -3 u + 2 v 2 -3 v -2 v 2 w + 4 v w -2 w + 1$ (db)
Jones polynomial	$q 3 -4 q 2 + 8 q -12 + 17 q -1 -18 q -2 + 19 q -3 -15 q -4 + 12 q -5 -6 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 8 + 3 z 2 a 6 + a 6 z -2 + 3 a 6 -3 z 4 a 4 -5 z 2 a 4 -2 a 4 z -2 -5 a 4 + z 6 a 2 + 2 z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2 -2 z 4 -2 z 2 + z 2 a -2$ (db)
Kauffman polynomial	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 7 a 3 z 9 + 4 a z 9 + 4 a 6 z 8 + 9 a 4 z 8 + 11 a 2 z 8 + 6 z 8 + 4 a 7 z 7 + 4 a 5 z 7 -5 a 3 z 7 - a z 7 + 4 z 7 a -1 + 3 a 8 z 6 + 2 a 6 z 6 -13 a 4 z 6 -27 a 2 z 6 + z 6 a -2 -14 z 6 + a 9 z 5 -3 a 7 z 5 -9 a 5 z 5 -8 a 3 z 5 -13 a z 5 -10 z 5 a -1 -6 a 8 z 4 -15 a 6 z 4 -4 a 4 z 4 + 15 a 2 z 4 -2 z 4 a -2 + 8 z 4 -2 a 9 z 3 -3 a 7 z 3 + a 5 z 3 + 4 a 3 z 3 + 8 a z 3 + 6 z 3 a -1 + 5 a 8 z 2 + 18 a 6 z 2 + 16 a 4 z 2 + a 2 z 2 + z 2 a -2 - z 2 + a 9 z + 3 a 7 z + 5 a 5 z + 3 a 3 z -2 a 8 -8 a 6 -9 a 4 -4 a 2 -2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

-2 is the signature of L11a416. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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