L11n300

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,13,19,14 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_14,10,15,9 X_20,16,21,15 X_10,19,5,20 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, 7, -9}, {11, -2, 3, -7, 8, 5, -4, -3, 9, -8, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 2 - v w u 2 + w u 2 - u 2 + v 2 u -3 v u - v 2 w u + 3 v w u - w u + u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$q 3 -3 q 2 + 4 q -5 + 7 q -1 -5 q -2 + 6 q -3 -3 q -4 + 2 q -5$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 6 z -2 + z 2 a 4 -2 a 4 z -2 -2 a 4 - z 4 a 2 + a 2 z -2 + 2 a 2 - z 4 - z 2 + z 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 4 a 2 z 8 + 3 z 8 -3 a 3 z 7 + 3 z 7 a -1 -4 a 4 z 6 -16 a 2 z 6 + z 6 a -2 -11 z 6 + a 5 z 5 + 3 a 3 z 5 -9 a z 5 -11 z 5 a -1 + 8 a 4 z 4 + 21 a 2 z 4 -3 z 4 a -2 + 10 z 4 + 8 a z 3 + 8 z 3 a -1 + 2 a 6 z 2 -2 a 4 z 2 -9 a 2 z 2 + z 2 a -2 -4 z 2 + 2 a 5 z + 2 a 3 z -2 a 6 -3 a 4 -2 a 2 -2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 L11n300. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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