L11a173

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_20,14,21,13 X_14,6,15,5 X_4,21,5,22 X_16,7,17,8 X_22,15,7,16 X_18,12,19,11 X_12,20,13,19 X_6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 9, -10, 4, -5, 8, -7, 11, -9, 10, -4, 6, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 2 v 2 u 5 -3 v u 5 + u 5 -3 v 2 u 4 + 5 v u 4 -2 u 4 + 3 v 2 u 3 -7 v u 3 + 3 u 3 -2 v 2 u 2 + 5 v u 2 -3 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 9 + a 3 z 7 -7 a z 7 + z 7 a -1 + 5 a 3 z 5 -18 a z 5 + 5 z 5 a -1 + 8 a 3 z 3 -20 a z 3 + 8 z 3 a -1 + 4 a 3 z -9 a z + 4 z a -1 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -5 a 3 z 9 -9 a z 9 -4 z 9 a -1 -6 a 4 z 8 - a 2 z 8 -4 z 8 a -2 + z 8 -5 a 5 z 7 + 13 a 3 z 7 + 32 a z 7 + 11 z 7 a -1 -3 z 7 a -3 -3 a 6 z 6 + 15 a 4 z 6 + 12 a 2 z 6 + 10 z 6 a -2 - z 6 a -4 + 5 z 6 - a 7 z 5 + 11 a 5 z 5 -17 a 3 z 5 -53 a z 5 -15 z 5 a -1 + 9 z 5 a -3 + 6 a 6 z 4 -15 a 4 z 4 -21 a 2 z 4 -5 z 4 a -2 + 3 z 4 a -4 -8 z 4 + 2 a 7 z 3 -7 a 5 z 3 + 7 a 3 z 3 + 39 a z 3 + 17 z 3 a -1 -6 z 3 a -3 - a 6 z 2 + 4 a 4 z 2 + 8 a 2 z 2 + z 2 a -2 -2 z 2 a -4 + 6 z 2 + 2 a 5 z -3 a 3 z -11 a z -6 z a -1 - a 2 + a 3 z -1 + a z -1$ (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 L11a173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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