L11a406

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 4 v u 2 -2 v w u 2 + 2 w u 2 -2 u 2 + 4 v 2 u -7 v u -2 v 2 w u + 7 v w u -4 w u + 2 u -2 v 2 + 2 v + 2 v 2 w -4 v w + 2 w$ (db)
Jones polynomial	$q -3 + 7 q -1 -10 q -2 + 15 q -3 -15 q -4 + 16 q -5 -13 q -6 + 10 q -7 -6 q -8 + 3 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 + 3 z 2 a 8 + 2 a 8 -2 z 4 a 6 - z 2 a 6 + a 6 z -2 -3 z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -4 a 4 - z 4 a 2 + 2 z 2 a 2 + a 2 z -2 + 3 a 2 + z 2$ (db)
Kauffman polynomial	$z 7 a 11 -4 z 5 a 11 + 5 z 3 a 11 -2 z a 11 + 3 z 8 a 10 -12 z 6 a 10 + 15 z 4 a 10 -7 z 2 a 10 + 2 a 10 + 3 z 9 a 9 -7 z 7 a 9 -3 z 5 a 9 + 12 z 3 a 9 -4 z a 9 + z 10 a 8 + 7 z 8 a 8 -34 z 6 a 8 + 42 z 4 a 8 -21 z 2 a 8 + 4 a 8 + 7 z 9 a 7 -15 z 7 a 7 + 3 z 5 a 7 + 2 z 3 a 7 + z 10 a 6 + 10 z 8 a 6 -30 z 6 a 6 + 23 z 4 a 6 -7 z 2 a 6 + a 6 z -2 -2 a 6 + 4 z 9 a 5 -8 z 5 a 5 -2 z 3 a 5 + 6 z a 5 -2 a 5 z -1 + 6 z 8 a 4 -2 z 6 a 4 -13 z 4 a 4 + 16 z 2 a 4 + 2 a 4 z -2 -7 a 4 + 7 z 7 a 3 -7 z 5 a 3 + z 3 a 3 + 4 z a 3 -2 a 3 z -1 + 6 z 6 a 2 -8 z 4 a 2 + 8 z 2 a 2 + a 2 z -2 -4 a 2 + 3 z 5 a -2 z 3 a + z 4 - z 2$ (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 L11a406. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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