L11n172

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_3,10,4,11 X_12,7,13,8 X_22,15,7,16 X_14,6,15,5 X_6,14,1,13 X_16,21,17,22 X_9,18,10,19 X_20,11,21,12 X_4,18,5,17 X_19,3,20,2
Gauss code	{1, 11, -2, -10, 5, -6}, {3, -1, -8, 2, 9, -3, 6, -5, 4, -7, 10, 8, -11, -9, 7, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 5 v u 2 -2 u 2 + 3 v 2 u -9 v u + 3 u -2 v 2 + 5 v -1$ (db)
Jones polynomial	$-2 q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 2 z 3 a 5 + z a 5 - a 5 z -1 - z 5 a 3 - z 3 a 3 + 2 a 3 z -1 + 3 z 3 a + 3 z a -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 5 z 9 -2 a 3 z 9 -4 a 6 z 8 -8 a 4 z 8 -4 a 2 z 8 -3 a 7 z 7 + a 5 z 7 + a 3 z 7 -3 a z 7 - a 8 z 6 + 12 a 6 z 6 + 25 a 4 z 6 + 11 a 2 z 6 - z 6 + 9 a 7 z 5 + 8 a 5 z 5 + 4 a 3 z 5 + 5 a z 5 + 3 a 8 z 4 -10 a 6 z 4 -29 a 4 z 4 -18 a 2 z 4 -2 z 4 -6 a 7 z 3 -5 a 5 z 3 -4 a 3 z 3 -8 a z 3 -3 z 3 a -1 -2 a 8 z 2 + 6 a 6 z 2 + 18 a 4 z 2 + 11 a 2 z 2 + z 2 + a 7 z -2 a 3 z + 3 a z + 4 z a -1 -2 a 6 -5 a 4 -3 a 2 + 1 + a 5 z -1 + 2 a 3 z -1 - a -1 z -1$ (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 =

-1 is the signature of L11n172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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