L11n245

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_10,11,1,12 X_14,5,15,6 X_9,19,10,18 X_17,3,18,2 X_16,8,17,7 X₃₈₄₉ X_20,16,21,15 X_22,13,11,14 X_4,20,5,19 X_6,21,7,22
Gauss code	{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + 2 v u 3 - u 3 -2 v 3 u 2 + 4 v 2 u 2 -5 v u 2 + 3 u 2 + 3 v 3 u -5 v 2 u + 4 v u -2 u - v 3 + 2 v 2 - v$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+11 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2 a z 5 - z 5 a -1 + 2 a 3 z 3 -5 a z 3 - z 3 a -1 + z 3 a -3 + 2 a 3 z -3 a z + z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -4 z 8 a -2 -9 z 8 -4 a 3 z 7 -3 a z 7 -2 z 7 a -1 -3 z 7 a -3 - a 4 z 6 + 10 a 2 z 6 + 10 z 6 a -2 - z 6 a -4 + 22 z 6 + 3 a 3 z 5 + 10 a z 5 + 16 z 5 a -1 + 9 z 5 a -3 -6 a 4 z 4 -14 a 2 z 4 -5 z 4 a -2 + 3 z 4 a -4 -16 z 4 -3 a 5 z 3 -2 a 3 z 3 -4 a z 3 -13 z 3 a -1 -8 z 3 a -3 + 5 a 4 z 2 + 8 a 2 z 2 -2 z 2 a -4 + 5 z 2 + a 5 z + 4 z a -1 + 3 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11n245. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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