L11a72

(Knotscape image)

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

Visit L11a72's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_10,19,11,20 X_22,15,5,16 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, 3, -9, 8, -4, 6, -5, 7, -3, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 7 -2 v u 6 + 2 u 6 + 3 v u 5 -3 u 5 -4 v u 4 + 4 u 4 + 4 v u 3 -4 u 3 -3 v u 2 + 3 u 2 + 2 v u -2 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{12}{q^{19/2}}+\frac{11}{q^{21/2}}-\frac{8}{q^{23/2}}+\frac{5}{q^{25/2}}-\frac{2}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -4 z a 13 -3 a 13 z -1 + 3 z 5 a 11 + 14 z 3 a 11 + 19 z a 11 + 7 a 11 z -1 -2 z 7 a 9 -11 z 5 a 9 -20 z 3 a 9 -15 z a 9 -4 a 9 z -1 - z 7 a 7 -5 z 5 a 7 -7 z 3 a 7 -2 z a 7$ (db)
Kauffman polynomial	$- z 4 a 18 + 2 z 2 a 18 - a 18 -2 z 5 a 17 + 2 z 3 a 17 -3 z 6 a 16 + 2 z 4 a 16 -4 z 7 a 15 + 5 z 5 a 15 -4 z 3 a 15 -4 z 8 a 14 + 6 z 6 a 14 -4 z 4 a 14 - z 2 a 14 -3 z 9 a 13 + 5 z 7 a 13 -6 z 5 a 13 + 11 z 3 a 13 -11 z a 13 + 3 a 13 z -1 - z 10 a 12 -4 z 8 a 12 + 20 z 6 a 12 -30 z 4 a 12 + 24 z 2 a 12 -7 a 12 -6 z 9 a 11 + 24 z 7 a 11 -44 z 5 a 11 + 53 z 3 a 11 -30 z a 11 + 7 a 11 z -1 - z 10 a 10 -2 z 8 a 10 + 19 z 6 a 10 -31 z 4 a 10 + 24 z 2 a 10 -7 a 10 -3 z 9 a 9 + 14 z 7 a 9 -26 z 5 a 9 + 29 z 3 a 9 -17 z a 9 + 4 a 9 z -1 -2 z 8 a 8 + 8 z 6 a 8 -8 z 4 a 8 + z 2 a 8 - z 7 a 7 + 5 z 5 a 7 -7 z 3 a 7 + 2 z a 7$ (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 =

-7 is the signature of L11a72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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