L11n74

(Knotscape image)

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

Visit L11n74'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_13,21,14,20 X_15,5,16,22 X_19,13,20,12 X_21,15,22,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u v^5+u v^4-u v^3+u v^2-u-v^7+v^5-v^4+v^3-v^2}{\sqrt{u} v^{7/2}}$ (db)
Jones polynomial	$q^{17/2}-q^{15/2}+2 q^{13/2}-3 q^{11/2}+3 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\sqrt{q}-\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 7 a -3 + z 5 a -1 -8 z 5 a -3 + 6 z 3 a -1 -20 z 3 a -3 + 2 z 3 a -5 + z 3 a -7 + 10 z a -1 -19 z a -3 + 5 z a -5 + 2 z a -7 + 4 a -1 z -1 -7 a -3 z -1 + 3 a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 - z 8 a -2 - z 8 a -4 + 9 z 7 a -1 + 10 z 7 a -3 - z 7 a -7 + 9 z 6 a -2 + 9 z 6 a -4 - z 6 a -6 - z 6 a -8 -28 z 5 a -1 -34 z 5 a -3 - z 5 a -5 + 4 z 5 a -7 - z 5 a -9 -25 z 4 a -2 -26 z 4 a -4 + 2 z 4 a -6 + 2 z 4 a -8 - z 4 a -10 + 37 z 3 a -1 + 49 z 3 a -3 + 3 z 3 a -5 -7 z 3 a -7 + 2 z 3 a -9 + 25 z 2 a -2 + 26 z 2 a -4 -3 z 2 a -6 - z 2 a -8 + 3 z 2 a -10 -21 z a -1 -30 z a -3 -7 z a -5 + 2 z a -7 -7 a -2 -7 a -4 - a -10 + 4 a -1 z -1 + 7 a -3 z -1 + 3 a -5 z -1$ (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 =

3 is the signature of L11n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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