L11a338

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_14,3,15,4 X_20,13,21,14 X_12,21,13,22 X_22,5,9,6 X_16,8,17,7 X_18,16,19,15 X_8,18,1,17 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 6, -10, 7, -9}, {10, -1, 2, -5, 4, -3, 8, -7, 9, -8, 11, -4, 5, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 u^3 v^3-3 u^3 v^2+u^3 v-4 u^2 v^3+12 u^2 v^2-9 u^2 v+2 u^2+2 u v^3-9 u v^2+12 u v-4 u+v^2-3 v+2}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$\frac{21}{q^{9/2}}-\frac{22}{q^{7/2}}+\frac{18}{q^{5/2}}+q^{3/2}-\frac{15}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{18}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 3 z 3 a 7 + 3 z a 7 - z 7 a 5 -4 z 5 a 5 -7 z 3 a 5 -4 z a 5 + a 5 z -1 - z 7 a 3 -3 z 5 a 3 -3 z 3 a 3 -2 z a 3 - a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$a 11 z 5 -2 a 11 z 3 + a 11 z + 3 a 10 z 6 -5 a 10 z 4 + 2 a 10 z 2 + 5 a 9 z 7 -6 a 9 z 5 + a 9 z 3 + 7 a 8 z 8 -12 a 8 z 6 + 13 a 8 z 4 -7 a 8 z 2 + 6 a 7 z 9 -7 a 7 z 7 + 4 a 7 z 5 + a 7 z + 2 a 6 z 10 + 12 a 6 z 8 -38 a 6 z 6 + 42 a 6 z 4 -15 a 6 z 2 + 12 a 5 z 9 -22 a 5 z 7 + 9 a 5 z 5 + 2 a 5 z 3 + a 5 z -1 + 2 a 4 z 10 + 12 a 4 z 8 -39 a 4 z 6 + 33 a 4 z 4 -9 a 4 z 2 - a 4 + 6 a 3 z 9 -6 a 3 z 7 -11 a 3 z 5 + 11 a 3 z 3 -4 a 3 z + a 3 z -1 + 7 a 2 z 8 -15 a 2 z 6 + 7 a 2 z 4 -2 a 2 z 2 + 4 a z 7 -9 a z 5 + 6 a z 3 -2 a z + z 6 -2 z 4 + z 2$ (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 L11a338. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$