L11a345

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_18,14,19,13 X_22,20,11,19 X_20,15,21,16 X_14,21,15,22 X_6,17,7,18 X_16,7,17,8 X_10,6,1,5 X_4,10,5,9 X_2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -7, 8, -2, 10, -9}, {11, -1, 3, -6, 5, -8, 7, -3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 4 v 2 u 3 -6 v u 3 + 2 u 3 + 2 v 3 u 2 -6 v 2 u 2 + 6 v u 2 -2 u 2 -2 v 3 u + 6 v 2 u -6 v u + 2 u + 2 v 3 -6 v 2 + 4 v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-19 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 10 a z 3 -8 z 3 a -1 + 2 z 3 a -3 + a 5 z -8 a 3 z + 12 a z -9 z a -1 + 2 z a -3 + 2 a 5 z -1 -5 a 3 z -1 + 6 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -7 a z 9 -5 z 9 a -1 -2 a 4 z 8 -6 a 2 z 8 -11 z 8 a -2 -15 z 8 - a 5 z 7 + a 3 z 7 + 6 a z 7 -9 z 7 a -1 -13 z 7 a -3 + 7 a 4 z 6 + 24 a 2 z 6 + 12 z 6 a -2 -9 z 6 a -4 + 38 z 6 + 5 a 5 z 5 + 16 a 3 z 5 + 30 a z 5 + 42 z 5 a -1 + 19 z 5 a -3 -4 z 5 a -5 -7 a 4 z 4 -21 a 2 z 4 + 5 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 -18 z 4 -9 a 5 z 3 -31 a 3 z 3 -50 a z 3 -40 z 3 a -1 -11 z 3 a -3 + z 3 a -5 + a 4 z 2 + 3 a 2 z 2 -9 z 2 a -2 -3 z 2 a -4 -4 z 2 + 7 a 5 z + 21 a 3 z + 29 a z + 19 z a -1 + 4 z a -3 + a 4 + a 2 + 3 a -2 + a -4 + 3-2 a 5 z -1 -5 a 3 z -1 -6 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L11a345. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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