L11n369

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_20,16,21,15 X_18,11,19,12 X_12,17,13,18 X_22,20,17,19 X_16,22,5,21 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -3 v u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 2 + 4 q -7 + 10 q -1 -10 q -2 + 12 q -3 -9 q -4 + 7 q -5 -3 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 6 + a 6 z -2 + a 6 -2 z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -4 a 4 + z 6 a 2 + 3 z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2 - z 4 - z 2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + a 6 z 8 + 5 a 4 z 8 + 4 a 2 z 8 + 6 a 3 z 7 + 6 a z 7 - a 6 z 6 -8 a 4 z 6 -3 a 2 z 6 + 4 z 6 + 3 a 7 z 5 + 3 a 5 z 5 -12 a 3 z 5 -11 a z 5 + z 5 a -1 + a 8 z 4 + 6 a 6 z 4 + 6 a 4 z 4 -6 a 2 z 4 -7 z 4 -3 a 7 z 3 -3 a 5 z 3 + 3 a 3 z 3 + 2 a z 3 - z 3 a -1 -2 a 8 z 2 -4 a 6 z 2 + a 4 z 2 + 5 a 2 z 2 + 2 z 2 + 4 a 5 z + 4 a 3 z + a 8 - a 6 -4 a 4 -3 a 2 -2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 L11n369. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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