L11n262

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_19,22,20,9 X_15,20,16,21 X_21,16,22,17 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 - v w u 3 - w u 3 + u 3 + 3 v u 2 + 3 w u 2 - u 2 -3 v u + v w u -3 w u + v - v w + w + 1$ (db)
Jones polynomial	$q -3 - q -5 + 5 q -6 -4 q -7 + 7 q -8 -6 q -9 + 6 q -10 -5 q -11 + 2 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 3 a 12 z -2 + 4 a 12 -5 z 2 a 10 -2 a 10 z -2 -7 a 10 + z 4 a 8 - a 8 z -2 - a 8 + z 6 a 6 + 6 z 4 a 6 + 8 z 2 a 6 + a 6 z -2 + 4 a 6$ (db)
Kauffman polynomial	$z 7 a 15 -5 z 5 a 15 + 9 z 3 a 15 -7 z a 15 + 2 a 15 z -1 + 2 z 8 a 14 -8 z 6 a 14 + 8 z 4 a 14 -2 z 2 a 14 - a 14 z -2 + a 14 + z 9 a 13 + 2 z 7 a 13 -24 z 5 a 13 + 39 z 3 a 13 -27 z a 13 + 8 a 13 z -1 + 6 z 8 a 12 -24 z 6 a 12 + 24 z 4 a 12 -10 z 2 a 12 -3 a 12 z -2 + 5 a 12 + z 9 a 11 + 4 z 7 a 11 -33 z 5 a 11 + 52 z 3 a 11 -34 z a 11 + 10 a 11 z -1 + 4 z 8 a 10 -18 z 6 a 10 + 24 z 4 a 10 -13 z 2 a 10 -2 a 10 z -2 + 4 a 10 + 3 z 7 a 9 -15 z 5 a 9 + 22 z 3 a 9 -10 z a 9 + 2 a 9 z -1 - z 6 a 8 + 2 z 4 a 8 + 3 z 2 a 8 + a 8 z -2 -3 a 8 - z 5 a 7 + 4 z a 7 -2 a 7 z -1 + z 6 a 6 -6 z 4 a 6 + 8 z 2 a 6 + a 6 z -2 -4 a 6$ (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 =

-4 is the signature of L11n262. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$	$i = -3$
$r = -11$		${\mathbb Z}$
$r = -10$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -7$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$