L11n175

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_14,5,15,6 X_18,12,19,11 X_3,10,4,11 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, -2, 5, 4, -6, 7, -3, 9, -7, 8, -4, -11, 2, 10, -8}

A Braid Representative

A Morse Link Presentation

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

-3 is the signature of L11n175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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