L11a312

(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_12,3,13,4 X_8,9,1,10 X_22,13,9,14 X_14,8,15,7 X_18,6,19,5 X_20,17,21,18 X_16,21,17,22 X_6,16,7,15 X_4,20,5,19
Gauss code	{1, -2, 3, -11, 7, -10, 6, -4}, {4, -1, 2, -3, 5, -6, 10, -9, 8, -7, 11, -8, 9, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)-2 t(1)-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-5 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + 2 z 5 a 5 + 6 z 3 a 5 + 3 z a 5 - z 7 a 3 -4 z 5 a 3 -5 z 3 a 3 -3 z a 3 + 2 z 5 a + 7 z 3 a + 5 z a + a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$a 10 z 4 - a 10 z 2 + 3 a 9 z 5 -3 a 9 z 3 + 5 a 8 z 6 -6 a 8 z 4 + 2 a 8 z 2 + 6 a 7 z 7 -10 a 7 z 5 + 9 a 7 z 3 -3 a 7 z + 5 a 6 z 8 -8 a 6 z 6 + 7 a 6 z 4 -3 a 6 z 2 + 3 a 5 z 9 -2 a 5 z 7 -5 a 5 z 5 + 6 a 5 z 3 -2 a 5 z + a 4 z 10 + 4 a 4 z 8 -16 a 4 z 6 + 15 a 4 z 4 -5 a 4 z 2 + 5 a 3 z 9 -14 a 3 z 7 + 10 a 3 z 5 -3 a 3 z 3 + a 2 z 10 + a 2 z 8 -11 a 2 z 6 + 9 a 2 z 4 + 2 a z 9 -5 a z 7 + z 7 a -1 -3 a z 5 -5 z 5 a -1 + 11 a z 3 + 8 z 3 a -1 -6 a z + a z -1 -5 z a -1 + a -1 z -1 + 2 z 8 -8 z 6 + 8 z 4 - z 2 -1$ (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 L11a312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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