L11n398

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_15,22,16,19 X_7,20,8,21 X_19,8,20,9 X_13,18,14,5 X_11,14,12,15 X_17,12,18,13 X_21,16,22,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, -3, 9, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 - v u 4 - v w u 4 + 2 u 4 + 2 v u 3 + w u 3 -2 u 3 - v u 2 + 2 v w u 2 -2 w u 2 -2 v w u + w u + u + v w$ (db)
Jones polynomial	$q -3 - q -5 + 4 q -6 -4 q -7 + 6 q -8 -5 q -9 + 6 q -10 -4 q -11 + 2 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 4 a 12 z -2 + 4 a 12 -5 z 2 a 10 -5 a 10 z -2 -9 a 10 + z 4 a 8 + z 2 a 8 + 2 a 8 z -2 + 2 a 8 + z 6 a 6 + 6 z 4 a 6 + 8 z 2 a 6 + 3 a 6$ (db)
Kauffman polynomial	$z 7 a 15 -5 z 5 a 15 + 8 z 3 a 15 -5 z a 15 + a 15 z -1 + 2 z 8 a 14 -9 z 6 a 14 + 12 z 4 a 14 -8 z 2 a 14 - a 14 z -2 + 4 a 14 + z 9 a 13 + z 7 a 13 -21 z 5 a 13 + 36 z 3 a 13 -23 z a 13 + 5 a 13 z -1 + 6 z 8 a 12 -28 z 6 a 12 + 41 z 4 a 12 -34 z 2 a 12 -4 a 12 z -2 + 18 a 12 + z 9 a 11 + 3 z 7 a 11 -32 z 5 a 11 + 57 z 3 a 11 -39 z a 11 + 9 a 11 z -1 + 4 z 8 a 10 -21 z 6 a 10 + 38 z 4 a 10 -37 z 2 a 10 -5 a 10 z -2 + 21 a 10 + 3 z 7 a 9 -17 z 5 a 9 + 30 z 3 a 9 -20 z a 9 + 5 a 9 z -1 - z 6 a 8 + 3 z 4 a 8 -3 z 2 a 8 -2 a 8 z -2 + 5 a 8 - z 5 a 7 + z 3 a 7 + z a 7 + z 6 a 6 -6 z 4 a 6 + 8 z 2 a 6 -3 a 6$ (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 =

-4 is the signature of L11n398. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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