L11a245

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_2,15,3,16 X_18,5,19,6 X_12,3,13,4 X_22,11,7,12 X_4,21,5,22 X_14,20,15,19 X_20,14,21,13 X_6,9,1,10 X_10,17,11,18
Gauss code	{1, -3, 5, -7, 4, -10}, {2, -1, 10, -11, 6, -5, 9, -8, 3, -2, 11, -4, 8, -9, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 4 + 2 u 4 -3 v 2 u 3 + 11 v u 3 -7 u 3 + 9 v 2 u 2 -17 v u 2 + 9 u 2 -7 v 2 u + 11 v u -3 u + 2 v 2 -2 v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{24}{q^{7/2}}+\frac{27}{q^{9/2}}-\frac{28}{q^{11/2}}+\frac{24}{q^{13/2}}-\frac{18}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 + z a 9 + z 5 a 7 -2 z 3 a 7 -2 z a 7 + a 7 z -1 + 3 z 5 a 5 + 4 z 3 a 5 + z a 5 - a 5 z -1 + z 5 a 3 -2 z 3 a 3 -3 z a 3 - z 3 a$ (db)
Kauffman polynomial	$- z 6 a 12 + z 4 a 12 -5 z 7 a 11 + 9 z 5 a 11 -3 z 3 a 11 - z a 11 -10 z 8 a 10 + 21 z 6 a 10 -11 z 4 a 10 + z 2 a 10 -10 z 9 a 9 + 16 z 7 a 9 -3 z 5 a 9 + z 3 a 9 -4 z 10 a 8 -13 z 8 a 8 + 45 z 6 a 8 -33 z 4 a 8 + 8 z 2 a 8 -21 z 9 a 7 + 43 z 7 a 7 -27 z 5 a 7 + 8 z 3 a 7 -3 z a 7 + a 7 z -1 -4 z 10 a 6 -16 z 8 a 6 + 50 z 6 a 6 -44 z 4 a 6 + 13 z 2 a 6 - a 6 -11 z 9 a 5 + 13 z 7 a 5 - z 5 a 5 -5 z 3 a 5 - z a 5 + a 5 z -1 -13 z 8 a 4 + 23 z 6 a 4 -19 z 4 a 4 + 6 z 2 a 4 -9 z 7 a 3 + 13 z 5 a 3 -8 z 3 a 3 + 3 z a 3 -4 z 6 a 2 + 4 z 4 a 2 - z 5 a + z 3 a$ (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 =

-3 is the signature of L11a245. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -6$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = -5$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -4$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$