L11n306

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_15,22,16,11 X_13,20,14,21 X_21,14,22,15 X_17,8,18,9 X_7,16,8,17 X_9,18,10,19 X_19,10,20,5 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 9}, {-11, 2, -4, 5, -3, 7, -6, 8, -9, 4, -5, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v u 3 + v 2 w u 3 - u 3 - v 2 w u 2 + u 2 + v 2 w u - v w u - u + v w - w + 1$ (db)
Jones polynomial	$q -1 - q -2 + 4 q -3 -3 q -4 + 5 q -5 -4 q -6 + 4 q -7 -3 q -8 + 2 q -9 - q -10$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 -2 a 10 + z 6 a 8 + 6 z 4 a 8 + 12 z 2 a 8 + 4 a 8 z -2 + 11 a 8 - z 8 a 6 -7 z 6 a 6 -18 z 4 a 6 -24 z 2 a 6 -5 a 6 z -2 -18 a 6 + z 6 a 4 + 6 z 4 a 4 + 12 z 2 a 4 + 2 a 4 z -2 + 9 a 4$ (db)
Kauffman polynomial	$z a 13 + 2 z 2 a 12 - a 12 + z 5 a 11 - z 3 a 11 -2 z a 11 + a 11 z -1 + 3 z 6 a 10 -10 z 4 a 10 + 6 z 2 a 10 - a 10 z -2 + 5 z 7 a 9 -23 z 5 a 9 + 31 z 3 a 9 -19 z a 9 + 5 a 9 z -1 + 4 z 8 a 8 -19 z 6 a 8 + 27 z 4 a 8 -21 z 2 a 8 -4 a 8 z -2 + 13 a 8 + z 9 a 7 + z 7 a 7 -25 z 5 a 7 + 50 z 3 a 7 -35 z a 7 + 9 a 7 z -1 + 5 z 8 a 6 -29 z 6 a 6 + 55 z 4 a 6 -46 z 2 a 6 -5 a 6 z -2 + 22 a 6 + z 9 a 5 -4 z 7 a 5 - z 5 a 5 + 18 z 3 a 5 -19 z a 5 + 5 a 5 z -1 + z 8 a 4 -7 z 6 a 4 + 18 z 4 a 4 -21 z 2 a 4 -2 a 4 z -2 + 11 a 4$ (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 =

-6 is the signature of L11n306. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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