L11n309

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,17,8,16 X_9,21,10,20 X_15,9,16,8 X_19,5,20,10 X_18,13,19,14 X_22,17,11,18 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, {11, -2, 7, 9, -5, 3, 8, -7, -6, 4, -9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 3 v u 2 - v w u 2 + w u 2 - u 2 + 3 v 2 u -4 v u - v 2 w u + 4 v w u -3 w u + u - v 2 + v + v 2 w -3 v w + w$ (db)
Jones polynomial	$- q 3 + 5 q 2 -6 q + 10-10 q -1 + 10 q -2 -8 q -3 + 6 q -4 -3 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -3 z 2 a 4 -2 a 4 + 2 z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2 + z 4 -2 z 2 -2 z -2 -4- z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 3 a 5 z 7 + 6 a 3 z 7 + 5 a z 7 + 2 z 7 a -1 + a 6 z 6 -5 a 4 z 6 -12 a 2 z 6 -6 z 6 -9 a 5 z 5 -24 a 3 z 5 -16 a z 5 - z 5 a -1 -3 a 6 z 4 -5 a 4 z 4 + a 2 z 4 + 5 z 4 a -2 + 8 z 4 + 7 a 5 z 3 + 20 a 3 z 3 + 13 a z 3 + z 3 a -1 + z 3 a -3 + 3 a 6 z 2 + 6 a 4 z 2 + 6 a 2 z 2 -2 z 2 a -2 + z 2 -2 a 5 z -6 a 3 z + 4 z a -1 - a 6 - a 4 -3 a 2 -4 a -2 -6-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11n309. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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