L11a478

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v w u 4 + v 2 u 3 -4 v u 3 - v 2 w u 3 + 4 v w u 3 -2 w u 3 + 2 u 3 -3 v 2 u 2 + 6 v u 2 + 3 v 2 w u 2 -6 v w u 2 + 3 w u 2 -3 u 2 + 2 v 2 u -4 v u -2 v 2 w u + 4 v w u - w u + u + v - v w$ (db)
Jones polynomial	$- q 2 + 4 q -8 + 13 q -1 -16 q -2 + 19 q -3 -17 q -4 + 15 q -5 -10 q -6 + 6 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 4 a 6 -5 z 2 a 6 -2 a 6 z -2 -6 a 6 + z 6 a 4 + 2 z 4 a 4 + 3 z 2 a 4 + a 4 z -2 + 3 a 4 + z 6 a 2 + 2 z 4 a 2 + 2 z 2 a 2 + a 2 - z 4 - z 2$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 5 z 2 a 10 -2 a 10 + 2 z 7 a 9 -5 z 5 a 9 + 2 z 3 a 9 + z a 9 + 3 z 8 a 8 -6 z 6 a 8 + 3 z 4 a 8 -3 z 2 a 8 - a 8 z -2 + 3 a 8 + 3 z 9 a 7 -4 z 7 a 7 + 4 z 3 a 7 -7 z a 7 + 2 a 7 z -1 + z 10 a 6 + 8 z 8 a 6 -27 z 6 a 6 + 37 z 4 a 6 -27 z 2 a 6 -2 a 6 z -2 + 11 a 6 + 7 z 9 a 5 -9 z 7 a 5 -2 z 5 a 5 + 12 z 3 a 5 -9 z a 5 + 2 a 5 z -1 + z 10 a 4 + 12 z 8 a 4 -32 z 6 a 4 + 36 z 4 a 4 -21 z 2 a 4 - a 4 z -2 + 7 a 4 + 4 z 9 a 3 + 4 z 7 a 3 -19 z 5 a 3 + 15 z 3 a 3 -2 z a 3 + 7 z 8 a 2 -8 z 6 a 2 + 7 z 7 a -11 z 5 a + 4 z 3 a - z a + 4 z 6 -6 z 4 + 2 z 2 + z 5 a -1 - z 3 a -1$ (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 L11a478. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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