L11n235

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_5,20,6,21 X_14,3,15,4 X_4,15,5,16 X_19,22,20,9 X_16,7,17,8 X_18,12,19,11 X_12,18,13,17 X_2,9,3,10 X_8,13,1,14 X_21,6,22,7
Gauss code	{1, -9, 3, -4, -2, 11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, -5, 2, -11, 5}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L11n235. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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