L11n75

(Knotscape image)

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

Visit L11n75's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 5 + u 4 -2 v u 3 + 2 u 3 + 2 v u 2 -2 u 2 + v u -2 v$ (db)
Jones polynomial	$q^{15/2}-q^{13/2}+2 q^{11/2}-2 q^{9/2}+q^{7/2}-q^{5/2}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 - z 5 a -3 + a z 3 -4 z 3 a -1 -6 z 3 a -3 + 3 a z - z a -1 -8 z a -3 + 3 z a -5 + z a -7 + a z -1 + a -1 z -1 -4 a -3 z -1 + 2 a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -2 z 8 a -2 - z 8 a -4 - z 8 - a z 7 + 6 z 7 a -1 + 8 z 7 a -3 - z 7 a -7 + 14 z 6 a -2 + 8 z 6 a -4 -2 z 6 a -6 - z 6 a -8 + 5 z 6 + 6 a z 5 -9 z 5 a -1 -21 z 5 a -3 -2 z 5 a -5 + 4 z 5 a -7 -27 z 4 a -2 -20 z 4 a -4 + 8 z 4 a -6 + 5 z 4 a -8 -4 z 4 -10 a z 3 + 5 z 3 a -1 + 27 z 3 a -3 + 9 z 3 a -5 -3 z 3 a -7 + 18 z 2 a -2 + 21 z 2 a -4 -6 z 2 a -6 -7 z 2 a -8 -2 z 2 + 6 a z -4 z a -1 -17 z a -3 -7 z a -5 -5 a -2 -6 a -4 + a -6 + 2 a -8 + 1- a z -1 + a -1 z -1 + 4 a -3 z -1 + 2 a -5 z -1$ (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 L11n75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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