L11n248

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,6,15,5 X_18,8,19,7 X_20,9,21,10 X_10,11,1,12 X_6,14,7,13 X_4,18,5,17 X_15,11,16,22 X_2,19,3,20 X_21,17,22,16
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, -9, 11, 8, -4, 10, -5, -11, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 -2 v u 3 + u 3 + 2 v 3 u 2 -6 v 2 u 2 + 6 v u 2 -2 u 2 -2 v 3 u + 6 v 2 u -6 v u + 2 u + v 3 -2 v 2 + 2 v -1$ (db)
Jones polynomial	$2 q^{11/2}-6 q^{9/2}+10 q^{7/2}-14 q^{5/2}+15 q^{3/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -4 z 5 a -1 + z 5 a -3 + 2 a z 3 -6 z 3 a -1 + z 3 a -3 + 2 a z -3 z a -1 + z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -1 -2 z 9 a -3 -10 z 8 a -2 -3 z 8 a -4 -7 z 8 -8 a z 7 -11 z 7 a -1 -4 z 7 a -3 - z 7 a -5 -4 a 2 z 6 + 14 z 6 a -2 + z 6 a -4 + 9 z 6 - a 3 z 5 + 15 a z 5 + 28 z 5 a -1 + 7 z 5 a -3 -5 z 5 a -5 + 5 a 2 z 4 -3 z 4 a -2 - z 4 a -4 -3 z 4 a -6 + a 3 z 3 -9 a z 3 -19 z 3 a -1 -2 z 3 a -3 + 7 z 3 a -5 - a 2 z 2 -2 z 2 a -2 + z 2 a -4 + 3 z 2 a -6 - z 2 + 4 a z + 6 z a -1 -2 z a -5 + 1- a z -1 - a -1 z -1$ (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 =

1 is the signature of L11n248. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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