L11n76

(Knotscape image)

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

Visit L11n76's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 5 v u 4 - u 4 -8 v u 3 + 6 u 3 + 6 v u 2 -8 u 2 - v u + 5 u -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+7 q^{5/2}-11 q^{3/2}+13 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 -2 z 5 a -1 -3 a 3 z 3 + 6 a z 3 -6 z 3 a -1 + z 3 a -3 + a 5 z -4 a 3 z + 5 a z -6 z a -1 + 2 z a -3 + a 5 z -1 -2 a 3 z -1 + 3 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -7 a 2 z 8 -4 z 8 a -2 -11 z 8 -8 a 3 z 7 -11 a z 7 -6 z 7 a -1 -3 z 7 a -3 -3 a 4 z 6 + 12 a 2 z 6 + 7 z 6 a -2 - z 6 a -4 + 23 z 6 + 16 a 3 z 5 + 35 a z 5 + 27 z 5 a -1 + 8 z 5 a -3 -3 a 4 z 4 -13 a 2 z 4 + 2 z 4 a -2 + 3 z 4 a -4 -11 z 4 -6 a 5 z 3 -21 a 3 z 3 -33 a z 3 -25 z 3 a -1 -7 z 3 a -3 + 3 a 4 z 2 + 9 a 2 z 2 -6 z 2 a -2 -3 z 2 a -4 + 3 z 2 + 5 a 5 z + 11 a 3 z + 15 a z + 12 z a -1 + 3 z a -3 -2 a 2 + 2 a -2 + a -4 - a 5 z -1 -2 a 3 z -1 -3 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11n76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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