L11a45

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_18,9,19,10 X_20,15,21,16 X_16,19,17,20 X_8,17,9,18 X_4,21,1,22 X_12,6,13,5 X_10,4,11,3 X_22,12,5,11 X_2,14,3,13
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -6, 3, -9, 10, -8, 11, -2, 4, -5, 6, -3, 5, -4, 7, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 6 v u 4 -6 u 4 -10 v u 3 + 10 u 3 + 10 v u 2 -10 u 2 -6 v u + 6 u + 2 v -2$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 3 a 3 z 5 + 3 a z 5 - z 5 a -1 -2 a 5 z 3 + 3 a 3 z 3 + 3 a z 3 -2 z 3 a -1 - a 5 z + 2 a 3 z - z a -1 - a 5 z -1 + 3 a 3 z -1 -2 a z -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -6 a 5 z 9 -12 a 3 z 9 -6 a z 9 -7 a 6 z 8 -12 a 4 z 8 -14 a 2 z 8 -9 z 8 -4 a 7 z 7 + 7 a 5 z 7 + 15 a 3 z 7 -4 a z 7 -8 z 7 a -1 - a 8 z 6 + 16 a 6 z 6 + 31 a 4 z 6 + 27 a 2 z 6 -4 z 6 a -2 + 9 z 6 + 9 a 7 z 5 + 2 a 5 z 5 + 20 a z 5 + 12 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -11 a 6 z 4 -19 a 4 z 4 -10 a 2 z 4 + 5 z 4 a -2 + z 4 -5 a 7 z 3 + a 3 z 3 -13 a z 3 -8 z 3 a -1 + z 3 a -3 - a 8 z 2 + 2 a 6 z 2 + 5 a 4 z 2 + a 2 z 2 -2 z 2 a -2 -3 z 2 -2 a 5 z -3 a 3 z + a z + 2 z a -1 - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 L11a45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$