L11n299

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,19,14,18 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_9,14,10,15 X_15,20,16,21 X_19,5,20,10 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 2 v 2 u 3 - v u 3 - u 3 - v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 + w u 2 + u 2 + v 2 w u + v w u -2 w u - v w + w$ (db)
Jones polynomial	$q 3 -2 q 2 + q + 3 q -2 -2 q -3 + 4 q -4 -3 q -5 + 3 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 + a 6 z -2 - a 6 + 2 z 4 a 4 + 6 z 2 a 4 -2 a 4 z -2 + a 4 - z 6 a 2 -5 z 4 a 2 -6 z 2 a 2 + a 2 z -2 - a 2 + z 2 + 1 + z 2 a -2$ (db)
Kauffman polynomial	$2 a 5 z 9 + 2 a 3 z 9 + 3 a 6 z 8 + 6 a 4 z 8 + 4 a 2 z 8 + z 8 + a 7 z 7 -8 a 5 z 7 -9 a 3 z 7 + 2 a z 7 + 2 z 7 a -1 -15 a 6 z 6 -35 a 4 z 6 -27 a 2 z 6 + z 6 a -2 -6 z 6 -4 a 7 z 5 + a 5 z 5 -15 a z 5 -10 z 5 a -1 + 19 a 6 z 4 + 55 a 4 z 4 + 48 a 2 z 4 -4 z 4 a -2 + 8 z 4 + 3 a 7 z 3 + 10 a 5 z 3 + 18 a 3 z 3 + 21 a z 3 + 10 z 3 a -1 -8 a 6 z 2 -29 a 4 z 2 -30 a 2 z 2 + 2 z 2 a -2 -7 z 2 - a 7 z -3 a 5 z -7 a 3 z -7 a z -2 z a -1 + 3 a 4 + 4 a 2 + 2-2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

0 is the signature of L11n299. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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