L11n304

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_22,19,11,20 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_14,21,15,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 -2 v 2 u 2 + 2 v u 2 + v 2 w u 2 -2 v w u 2 + 2 w u 2 - u 2 - v u + v w u -2 w u + u + w$ (db)
Jones polynomial	$2 q 2 -3 q + 6-6 q -1 + 8 q -2 -6 q -3 + 6 q -4 -4 q -5 + 2 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -3 a 6 + 3 z 4 a 4 + 12 z 2 a 4 + 4 a 4 z -2 + 13 a 4 -2 z 6 a 2 -11 z 4 a 2 -21 z 2 a 2 -5 a 2 z -2 -17 a 2 + 2 z 4 + 7 z 2 + 2 z -2 + 7$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 6 a 4 z 8 + 4 a 2 z 8 + a 7 z 7 + a 5 z 7 + 4 a 3 z 7 + 4 a z 7 -9 a 6 z 6 -26 a 4 z 6 -15 a 2 z 6 + 2 z 6 -5 a 7 z 5 -20 a 5 z 5 -28 a 3 z 5 -12 a z 5 + z 5 a -1 + 12 a 6 z 4 + 38 a 4 z 4 + 26 a 2 z 4 + 8 a 7 z 3 + 35 a 5 z 3 + 44 a 3 z 3 + 18 a z 3 + z 3 a -1 -7 a 6 z 2 -32 a 4 z 2 -32 a 2 z 2 + 3 z 2 a -2 -4 z 2 -5 a 7 z -21 a 5 z -33 a 3 z -16 a z + z a -1 + 4 a 6 + 17 a 4 + 20 a 2 -2 a -2 + 6 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 L11n304. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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