L11n67

(Knotscape image)

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

Visit L11n67's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_9,20,10,21 X_13,18,14,19 X_7,14,8,15 X_17,8,18,9 X_19,12,20,13 X_15,22,16,5 X_21,16,22,17 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -3, 2, -11, 7, -4, 5, -8, 9, -6, 4, -7, 3, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-5 v u 3 + u 3 + 9 v u 2 -6 u 2 -6 v u + 9 u + v -5$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{17/2}}+\frac{10}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{3}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + z 3 a 11 + 2 z a 11 + a 11 z -1 - z 5 a 9 + 4 z a 9 + 2 a 9 z -1 -3 z 5 a 7 -9 z 3 a 7 -7 z a 7 -2 a 7 z -1 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$-6 z 4 a 14 + 11 z 2 a 14 -4 a 14 -3 z 7 a 13 + 3 z 5 a 13 - z 3 a 13 - z a 13 + a 13 z -1 -5 z 8 a 12 + 13 z 6 a 12 -27 z 4 a 12 + 28 z 2 a 12 -9 a 12 -2 z 9 a 11 -5 z 7 a 11 + 14 z 5 a 11 -11 z 3 a 11 + z a 11 + a 11 z -1 -10 z 8 a 10 + 19 z 6 a 10 -17 z 4 a 10 + 11 z 2 a 10 -4 a 10 -2 z 9 a 9 -8 z 7 a 9 + 24 z 5 a 9 -24 z 3 a 9 + 12 z a 9 -2 a 9 z -1 -5 z 8 a 8 + 3 z 6 a 8 + 8 z 4 a 8 -7 z 2 a 8 + 2 a 8 -6 z 7 a 7 + 12 z 5 a 7 -12 z 3 a 7 + 9 z a 7 -2 a 7 z -1 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L11n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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