L11a391

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 - w u 5 + u 5 + 3 v u 4 -2 v w u 4 + 3 w u 4 -2 u 4 -3 v u 3 + 2 v w u 3 -3 w u 3 + 2 u 3 + 3 v u 2 -2 v w u 2 + 3 w u 2 -2 u 2 -3 v u + 2 v w u -3 w u + 2 u + v - v w + w$ (db)
Jones polynomial	$q -3 -3 q -4 + 7 q -5 -8 q -6 + 14 q -7 -14 q -8 + 15 q -9 -12 q -10 + 9 q -11 -6 q -12 + 2 q -13 - q -14$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-2 a 14 z -2 - a 14 + 4 z 2 a 12 + 7 a 12 z -2 + 12 a 12 -6 z 4 a 10 -23 z 2 a 10 -8 a 10 z -2 -24 a 10 + 3 z 6 a 8 + 14 z 4 a 8 + 21 z 2 a 8 + 3 a 8 z -2 + 13 a 8 + z 6 a 6 + 3 z 4 a 6 + z 2 a 6$ (db)
Kauffman polynomial	$z 5 a 17 -3 z 3 a 17 + 3 z a 17 - a 17 z -1 + 2 z 6 a 16 -3 z 4 a 16 + a 16 + 3 z 7 a 15 -3 z 5 a 15 -2 z 3 a 15 + 3 z a 15 - a 15 z -1 + 3 z 8 a 14 -2 z 6 a 14 + 2 z 4 a 14 -9 z 2 a 14 -2 a 14 z -2 + 7 a 14 + 2 z 9 a 13 + 3 z 7 a 13 -13 z 5 a 13 + 22 z 3 a 13 -21 z a 13 + 7 a 13 z -1 + z 10 a 12 + 5 z 8 a 12 -18 z 6 a 12 + 33 z 4 a 12 -35 z 2 a 12 -7 a 12 z -2 + 22 a 12 + 6 z 9 a 11 -11 z 7 a 11 -8 z 5 a 11 + 46 z 3 a 11 -45 z a 11 + 15 a 11 z -1 + z 10 a 10 + 8 z 8 a 10 -38 z 6 a 10 + 62 z 4 a 10 -51 z 2 a 10 -8 a 10 z -2 + 28 a 10 + 4 z 9 a 9 -8 z 7 a 9 -7 z 5 a 9 + 27 z 3 a 9 -24 z a 9 + 8 a 9 z -1 + 6 z 8 a 8 -23 z 6 a 8 + 31 z 4 a 8 -24 z 2 a 8 -3 a 8 z -2 + 13 a 8 + 3 z 7 a 7 -8 z 5 a 7 + 2 z 3 a 7 + z 6 a 6 -3 z 4 a 6 + z 2 a 6$ (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 =

-6 is the signature of L11a391. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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