L11n412

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_11,18,12,19 X_17,12,18,7 X_15,20,16,21 X_19,22,20,13 X_21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + 2 v 2 u 2 - v w 2 u 2 + 2 w 2 u 2 - v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 -2 v 2 u + v w 2 u -2 w 2 u + v u + v 2 w u -2 v w u + w u + w 2$ (db)
Jones polynomial	$2 q -1 -3 q -2 + 6 q -3 -6 q -4 + 8 q -5 -6 q -6 + 6 q -7 -4 q -8 + 2 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 z -2 - a 10 + 3 z 2 a 8 + 4 a 8 z -2 + 6 a 8 -2 z 4 a 6 -6 z 2 a 6 -5 a 6 z -2 -9 a 6 - z 4 a 4 + 2 a 4 z -2 + 2 a 4 + 2 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 7 a 11 -5 z 5 a 11 + 8 z 3 a 11 -5 z a 11 + a 11 z -1 + 2 z 8 a 10 -9 z 6 a 10 + 12 z 4 a 10 -7 z 2 a 10 - a 10 z -2 + 4 a 10 + z 9 a 9 + z 7 a 9 -20 z 5 a 9 + 35 z 3 a 9 -21 z a 9 + 5 a 9 z -1 + 6 z 8 a 8 -26 z 6 a 8 + 38 z 4 a 8 -32 z 2 a 8 -4 a 8 z -2 + 17 a 8 + z 9 a 7 + 4 z 7 a 7 -28 z 5 a 7 + 44 z 3 a 7 -33 z a 7 + 9 a 7 z -1 + 4 z 8 a 6 -15 z 6 a 6 + 26 z 4 a 6 -32 z 2 a 6 -5 a 6 z -2 + 20 a 6 + 4 z 7 a 5 -12 z 5 a 5 + 18 z 3 a 5 -16 z a 5 + 5 a 5 z -1 + 2 z 6 a 4 -4 z 2 a 4 -2 a 4 z -2 + 6 a 4 + z 5 a 3 + z 3 a 3 + z a 3 + 3 z 2 a 2 -2 a 2$ (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 L11n412. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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