L11a74

(Knotscape image)

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

Visit L11a74's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_22,13,5,14 X_20,18,21,17 X_18,9,19,10 X_8,19,9,20 X_14,21,15,22 X_10,16,11,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 6, -9, 11, -2, 4, -8, 9, -3, 5, -6, 7, -5, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 6 u 4 + 8 v u 3 -12 u 3 -12 v u 2 + 8 u 2 + 6 v u -2 u - v$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 -4 z a 7 -3 a 7 z -1 + 5 z 3 a 5 + 8 z a 5 + 4 a 5 z -1 -2 z 5 a 3 -4 z 3 a 3 -6 z a 3 -2 a 3 z -1 - z 5 a + z 3 a -1$ (db)
Kauffman polynomial	$- a 6 z 10 - a 4 z 10 -3 a 7 z 9 -8 a 5 z 9 -5 a 3 z 9 -3 a 8 z 8 -10 a 6 z 8 -17 a 4 z 8 -10 a 2 z 8 - a 9 z 7 + 4 a 7 z 7 + 9 a 5 z 7 -7 a 3 z 7 -11 a z 7 + 11 a 8 z 6 + 42 a 6 z 6 + 52 a 4 z 6 + 13 a 2 z 6 -8 z 6 + 4 a 9 z 5 + 14 a 7 z 5 + 30 a 5 z 5 + 39 a 3 z 5 + 15 a z 5 -4 z 5 a -1 -13 a 8 z 4 -46 a 6 z 4 -45 a 4 z 4 -4 a 2 z 4 - z 4 a -2 + 7 z 4 -6 a 9 z 3 -28 a 7 z 3 -50 a 5 z 3 -38 a 3 z 3 -8 a z 3 + 2 z 3 a -1 + 6 a 8 z 2 + 18 a 6 z 2 + 15 a 4 z 2 + 2 a 2 z 2 - z 2 + 4 a 9 z + 16 a 7 z + 25 a 5 z + 15 a 3 z + 2 a z - a 8 -3 a 6 -2 a 4 - a 2 - a 9 z -1 -3 a 7 z -1 -4 a 5 z -1 -2 a 3 z -1$ (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 L11a74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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