L11a75

(Knotscape image)

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

Visit L11a75's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 5 -3 v u 4 + 4 u 4 + 4 v u 3 -4 u 3 -4 v u 2 + 4 u 2 + 4 v u -3 u -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{11}{q^{17/2}}+\frac{9}{q^{19/2}}-\frac{7}{q^{21/2}}+\frac{5}{q^{23/2}}-\frac{2}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 -2 a 13 z -1 + 3 z 3 a 11 + 8 z a 11 + 4 a 11 z -1 -2 z 5 a 9 -6 z 3 a 9 -4 z a 9 - a 9 z -1 -2 z 5 a 7 -6 z 3 a 7 -4 z a 7 - a 7 z -1 - z 5 a 5 -3 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 4 z 4 a 16 -5 z 2 a 16 + 2 a 16 -2 z 7 a 15 + 6 z 5 a 15 -3 z 3 a 15 - z a 15 -2 z 8 a 14 + 3 z 6 a 14 + 5 z 4 a 14 -5 z 2 a 14 + a 14 -2 z 9 a 13 + 5 z 7 a 13 -7 z 5 a 13 + 13 z 3 a 13 -8 z a 13 + 2 a 13 z -1 - z 10 a 12 + 7 z 6 a 12 -17 z 4 a 12 + 19 z 2 a 12 -6 a 12 -5 z 9 a 11 + 21 z 7 a 11 -40 z 5 a 11 + 34 z 3 a 11 -15 z a 11 + 4 a 11 z -1 - z 10 a 10 - z 8 a 10 + 13 z 6 a 10 -29 z 4 a 10 + 18 z 2 a 10 -5 a 10 -3 z 9 a 9 + 11 z 7 a 9 -17 z 5 a 9 + 6 z 3 a 9 -2 z a 9 + a 9 z -1 -3 z 8 a 8 + 8 z 6 a 8 -6 z 4 a 8 -2 z 2 a 8 + a 8 -3 z 7 a 7 + 9 z 5 a 7 -9 z 3 a 7 + 5 z a 7 - a 7 z -1 -2 z 6 a 6 + 5 z 4 a 6 - z 2 a 6 - z 5 a 5 + 3 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L11a75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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