L11a12

(Knotscape image)

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

Visit L11a12's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,12,15,11 X₈₄₉₃ X_12,5,13,6 X_22,13,5,14 X_18,15,19,16 X_16,9,17,10 X_10,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -12 v u 3 + 12 u 3 + 12 v u 2 -12 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{21}{q^{5/2}}-\frac{25}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 -3 z 3 a 7 -4 z a 7 - a 7 z -1 + 3 z 5 a 5 + 7 z 3 a 5 + 6 z a 5 + 2 a 5 z -1 - z 7 a 3 -3 z 5 a 3 -4 z 3 a 3 -2 z a 3 + z 5 a + z 3 a - z a - a z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + z 3 a 11 -4 z 6 a 10 + 6 z 4 a 10 -3 z 2 a 10 -7 z 7 a 9 + 9 z 5 a 9 -5 z 3 a 9 + 2 z a 9 -8 z 8 a 8 + 5 z 6 a 8 + 7 z 4 a 8 -9 z 2 a 8 + 2 a 8 -6 z 9 a 7 -5 z 7 a 7 + 25 z 5 a 7 -24 z 3 a 7 + 8 z a 7 - a 7 z -1 -2 z 10 a 6 -17 z 8 a 6 + 36 z 6 a 6 -15 z 4 a 6 -8 z 2 a 6 + 5 a 6 -13 z 9 a 5 + 10 z 7 a 5 + 26 z 5 a 5 -31 z 3 a 5 + 11 z a 5 -2 a 5 z -1 -2 z 10 a 4 -18 z 8 a 4 + 47 z 6 a 4 -27 z 4 a 4 + 3 a 4 -7 z 9 a 3 + 3 z 7 a 3 + 21 z 5 a 3 -18 z 3 a 3 + 4 z a 3 -9 z 8 a 2 + 19 z 6 a 2 -10 z 4 a 2 + 2 z 2 a 2 - a 2 -5 z 7 a + 10 z 5 a -5 z 3 a - z a + a z -1 - z 6 + z 4$ (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 L11a12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$