L11a411

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -2 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 -4 v 2 u + 5 v u + 2 v 2 w u -5 v w u + 4 w u -2 u + 2 v 2 -4 v + 2 v w -2 w + 2$ (db)
Jones polynomial	$- q 8 + 3 q 7 -6 q 6 + 10 q 5 -13 q 4 + 15 q 3 -14 q 2 + 13 q -8 + 6 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 + z 2 a -2 + 4 z 2 a -4 -2 z 2 a -6 -5 z 2 + 2 a 2 + a -2 + 2 a -4 - a -6 -4 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 3 z 8 a -2 + 7 z 8 a -4 + 6 z 8 a -6 + 2 z 8 + 2 a z 7 - z 7 a -1 -15 z 7 a -3 -7 z 7 a -5 + 5 z 7 a -7 + a 2 z 6 -9 z 6 a -2 -27 z 6 a -4 -16 z 6 a -6 + 3 z 6 a -8 -5 a z 5 -2 z 5 a -1 + 18 z 5 a -3 + 3 z 5 a -5 -11 z 5 a -7 + z 5 a -9 -4 a 2 z 4 + 7 z 4 a -2 + 41 z 4 a -4 + 19 z 4 a -6 -6 z 4 a -8 -13 z 4 + a z 3 -6 z 3 a -1 -8 z 3 a -3 + 8 z 3 a -5 + 7 z 3 a -7 -2 z 3 a -9 + 6 a 2 z 2 -3 z 2 a -2 -22 z 2 a -4 -9 z 2 a -6 + z 2 a -8 + 15 z 2 + 4 a z + 6 z a -1 -4 z a -5 -2 z a -7 -4 a 2 -2 a -2 + 4 a -4 + 2 a -6 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11a411. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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