L11n305

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_19,11,20,22 X_10,15,5,16 X_8,17,9,18 X_16,7,17,8 X_18,9,19,10 X_21,15,22,14 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 - v w u 4 + v 2 u 3 -2 v u 3 + 2 v w u 3 - w u 3 + u 3 - v 2 u 2 + 2 v u 2 -2 v w u 2 + w u 2 + v 2 u -2 v u - v 2 w u + 2 v w u - w u + v - v w + w$ (db)
Jones polynomial	$-2 + 4 q -1 -6 q -2 + 9 q -3 -8 q -4 + 9 q -5 -6 q -6 + 5 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + 2 a 8 z -2 + 3 a 8 -3 z 4 a 6 -12 z 2 a 6 -5 a 6 z -2 -15 a 6 + 2 z 6 a 4 + 11 z 4 a 4 + 22 z 2 a 4 + 4 a 4 z -2 + 17 a 4 -2 z 4 a 2 -6 z 2 a 2 - a 2 z -2 -5 a 2$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 5 z 2 a 10 -2 a 10 + 2 z 7 a 9 -6 z 5 a 9 + 3 z 3 a 9 + z a 9 + 2 z 8 a 8 -4 z 6 a 8 -4 z 2 a 8 -2 a 8 z -2 + 6 a 8 + z 9 a 7 + 2 z 7 a 7 -14 z 5 a 7 + 19 z 3 a 7 -16 z a 7 + 5 a 7 z -1 + 6 z 8 a 6 -22 z 6 a 6 + 36 z 4 a 6 -36 z 2 a 6 -5 a 6 z -2 + 20 a 6 + z 9 a 5 + 4 z 7 a 5 -23 z 5 a 5 + 43 z 3 a 5 -33 z a 5 + 9 a 5 z -1 + 4 z 8 a 4 -16 z 6 a 4 + 34 z 4 a 4 -32 z 2 a 4 -4 a 4 z -2 + 17 a 4 + 4 z 7 a 3 -15 z 5 a 3 + 30 z 3 a 3 -21 z a 3 + 5 a 3 z -1 + z 6 a 2 + 2 z 4 a 2 -5 z 2 a 2 - a 2 z -2 + 4 a 2 + 3 z 3 a -5 z a + a z -1$ (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 L11n305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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