L11a380

(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 X_14,3,15,4 X_20,15,21,16 X_18,6,19,5 X_6,11,7,12 X_22,7,11,8 X_8,18,9,17 X_16,10,17,9 X_4,20,5,19 X_10,21,1,22
Gauss code	{1, -2, 3, -10, 5, -6, 7, -8, 9, -11}, {6, -1, 2, -3, 4, -9, 8, -5, 10, -4, 11, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 4 -2 v 2 u 4 + v u 4 -4 v 3 u 3 + 6 v 2 u 3 -3 v u 3 + u 3 - v 4 u 2 + 5 v 3 u 2 -7 v 2 u 2 + 5 v u 2 - u 2 + v 4 u -3 v 3 u + 6 v 2 u -4 v u + v 3 -2 v 2 + v$ (db)
Jones polynomial	$-q^{5/2}+3 q^{3/2}-7 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + 2 z 5 a 5 + 5 z 3 a 5 + 2 z a 5 - z 7 a 3 -3 z 5 a 3 -2 z 3 a 3 + a 3 z -1 + 2 z 5 a + 5 z 3 a + z a - a z -1 - z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 2 z 3 a 9 -6 z 6 a 8 + 6 z 4 a 8 -3 z 2 a 8 -9 z 7 a 7 + 15 z 5 a 7 -14 z 3 a 7 + 5 z a 7 -9 z 8 a 6 + 15 z 6 a 6 -9 z 4 a 6 + z 2 a 6 -6 z 9 a 5 + 5 z 7 a 5 + 11 z 5 a 5 -14 z 3 a 5 + 5 z a 5 -2 z 10 a 4 -8 z 8 a 4 + 31 z 6 a 4 -25 z 4 a 4 + 7 z 2 a 4 -10 z 9 a 3 + 27 z 7 a 3 -19 z 5 a 3 + 8 z 3 a 3 -4 z a 3 + a 3 z -1 -2 z 10 a 2 -2 z 8 a 2 + 21 z 6 a 2 -21 z 4 a 2 + 6 z 2 a 2 - a 2 -4 z 9 a + 12 z 7 a -8 z 5 a + z 3 a -2 z a + a z -1 -3 z 8 + 11 z 6 -12 z 4 + 4 z 2 - z 7 a -1 + 4 z 5 a -1 -5 z 3 a -1 + 2 z a -1$ (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 =

-3 is the signature of L11a380. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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