L11a419

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -3 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 + u 2 -4 v 2 u + 6 v u + 3 v 2 w u -6 v w u + 4 w u -3 u + 2 v 2 -4 v - v 2 w + 3 v w -2 w + 2$ (db)
Jones polynomial	$- q 8 + 4 q 7 -8 q 6 + 12 q 5 -16 q 4 + 18 q 3 -16 q 2 + 15 q -9 + 6 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 2 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 + 3 z 2 a -2 + z 2 a -4 - z 2 a -6 -5 z 2 + 2 a 2 + 4 a -2 - a -4 -5 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 4 z 8 a -2 + 9 z 8 a -4 + 7 z 8 a -6 + 2 z 8 + 2 a z 7 + z 7 a -1 -7 z 7 a -3 + z 7 a -5 + 7 z 7 a -7 + a 2 z 6 -4 z 6 a -2 -18 z 6 a -4 -10 z 6 a -6 + 4 z 6 a -8 + z 6 -5 a z 5 -6 z 5 a -1 + 3 z 5 a -3 -9 z 5 a -5 -12 z 5 a -7 + z 5 a -9 -4 a 2 z 4 -10 z 4 a -2 + 11 z 4 a -4 + 4 z 4 a -6 -6 z 4 a -8 -15 z 4 + 2 a z 3 - z 3 a -1 -4 z 3 a -3 + 5 z 3 a -5 + 5 z 3 a -7 - z 3 a -9 + 6 a 2 z 2 + 15 z 2 a -2 - z 2 a -6 + z 2 a -8 + 19 z 2 + 3 a z + 5 z a -1 + 3 z a -3 + z a -5 -4 a 2 -8 a -2 -2 a -4 -9-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11a419. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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