L11n45

(Knotscape image)

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

Visit L11n45's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,14,6,15 X₃₈₄₉ X_9,18,10,19 X_11,20,12,21 X_13,22,14,5 X_19,10,20,11 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, -9, 7, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + v u 4 - u 4 - v u 3 + u 3 + v u 2 - u 2 - v u + u + v -1$ (db)
Jones polynomial	$-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -5 z 3 a 9 -7 z a 9 -3 a 9 z -1 + z 7 a 7 + 7 z 5 a 7 + 18 z 3 a 7 + 20 z a 7 + 7 a 7 z -1 -2 z 5 a 5 -10 z 3 a 5 -13 z a 5 -4 a 5 z -1$ (db)
Kauffman polynomial	$- z 4 a 14 + 3 z 2 a 14 - a 14 - z 5 a 13 + 2 z 3 a 13 - z 6 a 12 + 2 z 4 a 12 - z 2 a 12 - z 7 a 11 + 3 z 5 a 11 -4 z 3 a 11 - z 8 a 10 + 4 z 6 a 10 -6 z 4 a 10 - z 9 a 9 + 6 z 7 a 9 -15 z 5 a 9 + 16 z 3 a 9 -10 z a 9 + 3 a 9 z -1 -2 z 8 a 8 + 11 z 6 a 8 -24 z 4 a 8 + 23 z 2 a 8 -7 a 8 - z 9 a 7 + 7 z 7 a 7 -22 z 5 a 7 + 36 z 3 a 7 -26 z a 7 + 7 a 7 z -1 - z 8 a 6 + 6 z 6 a 6 -15 z 4 a 6 + 19 z 2 a 6 -7 a 6 -3 z 5 a 5 + 14 z 3 a 5 -16 z a 5 + 4 a 5 z -1$ (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 =

-5 is the signature of L11n45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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