L11a462

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,5,17,6 X_18,14,19,13 X_12,15,5,16 X_10,19,11,20 X_8,21,9,22 X_20,7,21,8 X_22,9,13,10 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 6 v 2 u 2 -4 v u 2 + 2 v 3 w u 2 -6 v 2 w u 2 + 5 v w u 2 - w u 2 + v 3 u -5 v 2 u + 6 v u + 4 v 2 w u -6 v w u + 2 w u -2 u + v 2 -2 v + 2 v w - w + 1$ (db)
Jones polynomial	$- q + 4-7 q -1 + 13 q -2 -17 q -3 + 21 q -4 -21 q -5 + 19 q -6 -14 q -7 + 10 q -8 -4 q -9 + q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 4 a 8 + 2 z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 6 a 6 -7 z 4 a 6 -8 z 2 a 6 -2 a 6 z -2 -5 a 6 + z 8 a 4 + 5 z 6 a 4 + 9 z 4 a 4 + 6 z 2 a 4 + a 4 z -2 + a 4 - z 6 a 2 -3 z 4 a 2 - z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 4 a 12 + 4 z 5 a 11 + 10 z 6 a 10 -10 z 4 a 10 + 6 z 2 a 10 -2 a 10 + 14 z 7 a 9 -18 z 5 a 9 + 6 z 3 a 9 + z a 9 + 13 z 8 a 8 -17 z 6 a 8 + 6 z 4 a 8 -6 z 2 a 8 - a 8 z -2 + 3 a 8 + 7 z 9 a 7 + 4 z 7 a 7 -32 z 5 a 7 + 22 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + 2 z 10 a 6 + 16 z 8 a 6 -50 z 6 a 6 + 42 z 4 a 6 -22 z 2 a 6 -2 a 6 z -2 + 9 a 6 + 12 z 9 a 5 -26 z 7 a 5 + 4 z 5 a 5 + 14 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + 2 z 10 a 4 + 7 z 8 a 4 -38 z 6 a 4 + 42 z 4 a 4 -14 z 2 a 4 - a 4 z -2 + 3 a 4 + 5 z 9 a 3 -15 z 7 a 3 + 11 z 5 a 3 + z a 3 + 4 z 8 a 2 -15 z 6 a 2 + 17 z 4 a 2 -4 z 2 a 2 -2 a 2 + z 7 a -3 z 5 a + 2 z 3 a$ (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 L11a462. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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