L11n251

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X₈₃₉₄ X_11,17,12,16 X_14,8,15,7 X_6,16,7,15 X_17,11,18,22 X_4,20,5,19 X_18,6,19,5 X_20,9,21,10 X_10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 -2 v u 3 + u 3 + v 3 u 2 -5 v 2 u 2 + 5 v u 2 - u 2 - v 3 u + 5 v 2 u -5 v u + u + v 3 -2 v 2 + 2 v -1$ (db)
Jones polynomial	$-2 q^{7/2}+5 q^{5/2}-8 q^{3/2}+11 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -3 a 3 z 3 + 10 a z 3 -2 z 3 a -1 -4 a 3 z + 7 a z -2 z a -1 - z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -5 a 2 z 8 - z 8 a -2 -6 z 8 -8 a 3 z 7 -11 a z 7 -3 z 7 a -1 -6 a 4 z 6 + 3 a 2 z 6 -2 z 6 a -2 + 7 z 6 -3 a 5 z 5 + 15 a 3 z 5 + 30 a z 5 + 9 z 5 a -1 -3 z 5 a -3 - a 6 z 4 + 7 a 4 z 4 + 6 a 2 z 4 + 9 z 4 a -2 + 7 z 4 + 2 a 5 z 3 -16 a 3 z 3 -29 a z 3 -5 z 3 a -1 + 6 z 3 a -3 + a 6 z 2 -4 a 4 z 2 -7 a 2 z 2 -6 z 2 a -2 -8 z 2 + 8 a 3 z + 14 a z + 4 z a -1 -2 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11n251. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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