L11n71

(Knotscape image)

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

Visit L11n71's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_17,22,18,5 X_11,18,12,19 X_21,12,22,13 X_13,20,14,21 X_19,14,20,15 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 -2 u 3 + v u 2 + 2 u 2 + 2 v u + u -2 v -2$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + 2 z a 11 + a 11 z -1 + 3 z a 9 + 2 a 9 z -1 - z 5 a 7 -5 z 3 a 7 -5 z a 7 -2 a 7 z -1 - z 5 a 5 -4 z 3 a 5 -2 z a 5$ (db)
Kauffman polynomial	$- z 8 a 14 + 7 z 6 a 14 -16 z 4 a 14 + 14 z 2 a 14 -4 a 14 - z 9 a 13 + 6 z 7 a 13 -10 z 5 a 13 + 5 z 3 a 13 - z a 13 + a 13 z -1 -3 z 8 a 12 + 20 z 6 a 12 -41 z 4 a 12 + 33 z 2 a 12 -9 a 12 - z 9 a 11 + 5 z 7 a 11 -3 z 5 a 11 -5 z 3 a 11 + 2 z a 11 + a 11 z -1 -2 z 8 a 10 + 13 z 6 a 10 -22 z 4 a 10 + 13 z 2 a 10 -4 a 10 -2 z 7 a 9 + 14 z 5 a 9 -24 z 3 a 9 + 13 z a 9 -2 a 9 z -1 - z 6 a 8 + 7 z 4 a 8 -8 z 2 a 8 + 2 a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 8 z a 7 -2 a 7 z -1 - z 6 a 6 + 4 z 4 a 6 -2 z 2 a 6 - z 5 a 5 + 4 z 3 a 5 -2 z a 5$ (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 =

-3 is the signature of L11n71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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