L10a173

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + v x u 2 - v w x u 2 + w x u 2 - x u 2 + u 2 -2 v 2 u + 4 v u + 2 v 2 w u -3 v w u + w u + v 2 x u -3 v x u - v 2 w x u + 4 v w x u -2 w x u + 2 x u - u + v 2 - v - v 2 w + v w - v 2 x + 2 v x + v 2 w x -2 v w x - x$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{17}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 + a 9 z -3 -3 z 3 a 7 -5 z a 7 -6 a 7 z -1 -3 a 7 z -3 + 2 z 5 a 5 + 5 z 3 a 5 + 9 z a 5 + 9 a 5 z -1 + 3 a 5 z -3 + z 5 a 3 - z 3 a 3 -5 z a 3 -4 a 3 z -1 - a 3 z -3 - z 3 a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 4 z 4 a 10 - z 2 a 10 -6 z 7 a 9 + 12 z 5 a 9 -14 z 3 a 9 + 12 z a 9 -5 a 9 z -1 + a 9 z -3 -5 z 8 a 8 + z 6 a 8 + 13 z 4 a 8 -17 z 2 a 8 -3 a 8 z -2 + 10 a 8 -2 z 9 a 7 -12 z 7 a 7 + 36 z 5 a 7 -42 z 3 a 7 + 29 z a 7 -12 a 7 z -1 + 3 a 7 z -3 -12 z 8 a 6 + 17 z 6 a 6 + 4 z 4 a 6 -23 z 2 a 6 -6 a 6 z -2 + 19 a 6 -2 z 9 a 5 -14 z 7 a 5 + 40 z 5 a 5 -39 z 3 a 5 + 23 z a 5 -12 a 5 z -1 + 3 a 5 z -3 -7 z 8 a 4 + 9 z 6 a 4 -7 z 2 a 4 -3 a 4 z -2 + 10 a 4 -8 z 7 a 3 + 16 z 5 a 3 -12 z 3 a 3 + 7 z a 3 -5 a 3 z -1 + a 3 z -3 -4 z 6 a 2 + 5 z 4 a 2 - z 5 a + z 3 a$ (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 L10a173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{3}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$