L11n165

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_15,20,16,21 X_14,5,15,6 X_4,13,5,14 X_17,22,18,7 X_21,16,22,17 X_19,12,20,13 X_11,18,12,19 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -9, 8, 5, -4, -3, 7, -6, 9, -8, 3, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 6 - v 2 u 5 + v u 5 - u 4 + v 2 u 3 - v u 3 + u 3 - v 2 u 2 + v u - u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2 z a 13 -2 a 13 z -1 + z 5 a 11 + 7 z 3 a 11 + 11 z a 11 + 5 a 11 z -1 - z 7 a 9 -6 z 5 a 9 -10 z 3 a 9 -7 z a 9 -3 a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -5 z a 7$ (db)
Kauffman polynomial	$- z 6 a 16 + 5 z 4 a 16 -6 z 2 a 16 + a 16 - z 7 a 15 + 5 z 5 a 15 -6 z 3 a 15 + 2 z a 15 - z 6 a 14 + 5 z 4 a 14 -4 z 2 a 14 + 3 z 3 a 13 -4 z a 13 + 2 a 13 z -1 - z 8 a 12 + 8 z 6 a 12 -19 z 4 a 12 + 19 z 2 a 12 -5 a 12 - z 9 a 11 + 8 z 7 a 11 -22 z 5 a 11 + 31 z 3 a 11 -21 z a 11 + 5 a 11 z -1 -2 z 8 a 10 + 13 z 6 a 10 -24 z 4 a 10 + 17 z 2 a 10 -5 a 10 - z 9 a 9 + 6 z 7 a 9 -11 z 5 a 9 + 12 z 3 a 9 -10 z a 9 + 3 a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -5 z 4 a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 5 z a 7$ (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 L11n165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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