L10a172

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_20,15,17,16 X_14,8,15,7 X_10,12,5,11 X_16,19,11,20 X_8,18,9,17 X_18,10,19,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {7, -8, 6, -3}, {9, -1, 4, -7, 8, -5}, {5, -2, 10, -4, 3, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v 2 w u 2 + v 2 x u 2 -2 v x u 2 - v 2 w x u 2 + v w x u 2 + x u 2 + v 2 u -2 v u -2 v 2 w u + v w u + v x u + v 2 w x u -2 v w x u + w x u -2 x u + u + v + v 2 w -2 v w + w + v w x - w x + x -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 q^{9/2}-11 q^{7/2}+9 q^{5/2}-11 q^{3/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 + 3 a z -11 z a -1 + 11 z a -3 -3 z a -5 + 3 a z -1 -8 a -1 z -1 + 7 a -3 z -1 -2 a -5 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -7 z 8 a -2 -5 z 8 a -4 -2 z 8 - a z 7 -4 z 7 a -1 -12 z 7 a -3 -9 z 7 a -5 + 19 z 6 a -2 + 4 z 6 a -4 -8 z 6 a -6 + 7 z 6 + 5 a z 5 + 30 z 5 a -1 + 51 z 5 a -3 + 20 z 5 a -5 -6 z 5 a -7 + 17 z 4 a -4 + 10 z 4 a -6 -3 z 4 a -8 -4 z 4 -10 a z 3 -48 z 3 a -1 -61 z 3 a -3 -17 z 3 a -5 + 5 z 3 a -7 - z 3 a -9 -26 z 2 a -2 -20 z 2 a -4 -2 z 2 a -6 -8 z 2 + 10 a z + 31 z a -1 + 35 z a -3 + 12 z a -5 -2 z a -7 + 19 a -2 + 10 a -4 + 10-5 a z -1 -12 a -1 z -1 -12 a -3 z -1 -5 a -5 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 L10a172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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