L11n253

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_14,3,15,4 X_9,18,10,19 X_5,16,6,17 X_22,7,11,8 X_6,21,7,22 X_20,15,21,16 X_17,8,18,9 X_19,4,20,5 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, {10, -1, 11, -2, 7, 4, -8, 3, -9, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 - v 4 u 3 + v 3 u 3 - v u 3 - v 2 u 2 - v 3 u + v u - u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 + z 5 a 11 + 6 z 3 a 11 + 7 z a 11 + a 11 z -1 - z 7 a 9 -6 z 5 a 9 -10 z 3 a 9 -6 z a 9 - a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -5 z a 7$ (db)
Kauffman polynomial	$- z 5 a 15 + 5 z 3 a 15 -5 z a 15 + z 2 a 14 + 2 z 3 a 13 -2 z a 13 - z 8 a 12 + 7 z 6 a 12 -14 z 4 a 12 + 10 z 2 a 12 - z 9 a 11 + 7 z 7 a 11 -16 z 5 a 11 + 18 z 3 a 11 -9 z a 11 + a 11 z -1 -2 z 8 a 10 + 12 z 6 a 10 -19 z 4 a 10 + 9 z 2 a 10 - a 10 - z 9 a 9 + 6 z 7 a 9 -11 z 5 a 9 + 11 z 3 a 9 -7 z a 9 + a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -5 z 4 a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 5 z a 7$ (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 =

-7 is the signature of L11n253. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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