L11n174

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_11,18,12,19 X_3,10,4,11 X_7,13,8,12 X_16,13,17,14 X_17,7,18,22 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v 2 u 3 - v u 3 - v 2 u 2 + 3 v u 2 - u 2 - v u + 3 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - z 5 a 7 -4 z 3 a 7 -4 z a 7 -2 a 7 z -1 + z 7 a 5 + 5 z 5 a 5 + 7 z 3 a 5 + 4 z a 5 + 2 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -4 z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$-2 z a 11 - z 4 a 10 - z 7 a 9 + 3 z 5 a 9 -4 z 3 a 9 + 3 z a 9 - a 9 z -1 -2 z 8 a 8 + 8 z 6 a 8 -10 z 4 a 8 + 6 z 2 a 8 - z 9 a 7 + z 7 a 7 + 9 z 5 a 7 -15 z 3 a 7 + 9 z a 7 -2 a 7 z -1 -4 z 8 a 6 + 17 z 6 a 6 -19 z 4 a 6 + 8 z 2 a 6 - a 6 - z 9 a 5 + z 7 a 5 + 11 z 5 a 5 -19 z 3 a 5 + 9 z a 5 -2 a 5 z -1 -2 z 8 a 4 + 9 z 6 a 4 -10 z 4 a 4 + 2 z 2 a 4 - z 7 a 3 + 5 z 5 a 3 -8 z 3 a 3 + 5 z a 3 - a 3 z -1$ (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 =

-5 is the signature of L11n174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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