L11n249

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_9,19,10,18 X_5,14,6,15 X_11,6,12,7 X_15,11,16,22 X_7,20,8,21 X₃₉₄₈ X_21,17,22,16 X_17,4,18,5 X_10,13,1,14 X_19,3,20,2
Gauss code	{1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -2 v u 3 + 2 v 3 u 2 -7 v 2 u 2 + 7 v u 2 - u 2 - v 3 u + 7 v 2 u -7 v u + 2 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 6 a z 3 -5 z 3 a -1 + z 3 a -3 + a 5 z -4 a 3 z + 3 a z -3 z a -1 + z a -3 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$-3 a z 9 -3 z 9 a -1 -8 a 2 z 8 -6 z 8 a -2 -14 z 8 -8 a 3 z 7 -8 a z 7 -4 z 7 a -1 -4 z 7 a -3 -3 a 4 z 6 + 13 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 31 z 6 + 13 a 3 z 5 + 27 a z 5 + 24 z 5 a -1 + 10 z 5 a -3 -3 a 4 z 4 -10 a 2 z 4 -7 z 4 a -2 + 2 z 4 a -4 -16 z 4 -6 a 5 z 3 -15 a 3 z 3 -18 a z 3 -16 z 3 a -1 -7 z 3 a -3 + a 4 z 2 + 3 a 2 z 2 - z 2 a -4 + 3 z 2 + 5 a 5 z + 7 a 3 z + 4 a z + 3 z a -1 + z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11n249. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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