L11n109

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,18,8,19 X_19,22,20,5 X_13,20,14,21 X_21,14,22,15 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 - u 3 + v u 2 + u 2 + v u + u - v -2$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + 2 z a 11 + 2 a 11 z -1 + z a 9 - a 9 z -1 - z 5 a 7 -4 z 3 a 7 -2 z a 7 + a 7 z -1 - z 5 a 5 -4 z 3 a 5 -3 z a 5 - a 5 z -1$ (db)
Kauffman polynomial	$- z 8 a 14 + 7 z 6 a 14 -15 z 4 a 14 + 11 z 2 a 14 -3 a 14 - z 9 a 13 + 7 z 7 a 13 -15 z 5 a 13 + 12 z 3 a 13 -4 z a 13 + a 13 z -1 -2 z 8 a 12 + 15 z 6 a 12 -34 z 4 a 12 + 28 z 2 a 12 -7 a 12 - z 9 a 11 + 8 z 7 a 11 -20 z 5 a 11 + 21 z 3 a 11 -10 z a 11 + 2 a 11 z -1 - z 8 a 10 + 8 z 6 a 10 -18 z 4 a 10 + 14 z 2 a 10 -4 a 10 + 2 z 3 a 9 -4 z a 9 + a 9 z -1 - z 6 a 8 + 4 z 4 a 8 -3 z 2 a 8 - z 7 a 7 + 4 z 5 a 7 -3 z 3 a 7 - z a 7 + a 7 z -1 - z 6 a 6 + 3 z 4 a 6 - a 6 - z 5 a 5 + 4 z 3 a 5 -3 z a 5 + a 5 z -1$ (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 L11n109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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