L10a174

(Knotscape image)

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

L10a174 is a closed five-link chain.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + v w u - w u + v x u - v w x u + 2 w x u -2 x u + 2 v y u - v w y u + w y u - v x y u - w x y u + 2 x y u -2 y u + u + v -2 v w + w - v x + 2 v w x -2 w x + x -2 v y + 2 v w y - w y + v x y - v w x y + w x y - x y + y$ (db)
Jones polynomial	$q -2 -4 q -3 + 10 q -4 -10 q -5 + 15 q -6 -11 q -7 + 15 q -8 -6 q -9 + 6 q -10 - q -11 + q -12$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 14 z -4 -5 a 12 z -2 -4 a 12 z -4 + 15 a 10 z -2 + 6 a 10 z -4 + 10 a 10 -10 z 2 a 8 -15 a 8 z -2 -4 a 8 z -4 -20 a 8 + 4 z 4 a 6 + 10 z 2 a 6 + 5 a 6 z -2 + a 6 z -4 + 10 a 6 + z 4 a 4$ (db)
Kauffman polynomial	$z 6 a 14 -5 z 4 a 14 + 10 z 2 a 14 + 5 a 14 z -2 - a 14 z -4 -10 a 14 + z 7 a 13 -10 z 3 a 13 + 20 z a 13 -15 a 13 z -1 + 4 a 13 z -3 + z 8 a 12 + 4 z 6 a 12 -20 z 4 a 12 + 30 z 2 a 12 + 14 a 12 z -2 -4 a 12 z -4 -25 a 12 + z 9 a 11 + 2 z 7 a 11 + 2 z 5 a 11 -30 z 3 a 11 + 55 z a 11 -41 a 11 z -1 + 12 a 11 z -3 + 6 z 8 a 10 -2 z 6 a 10 -25 z 4 a 10 + 40 z 2 a 10 + 18 a 10 z -2 -6 a 10 z -4 -31 a 10 + z 9 a 9 + 11 z 7 a 9 -12 z 5 a 9 -30 z 3 a 9 + 55 z a 9 -41 a 9 z -1 + 12 a 9 z -3 + 5 z 8 a 8 + 5 z 6 a 8 -25 z 4 a 8 + 30 z 2 a 8 + 14 a 8 z -2 -4 a 8 z -4 -25 a 8 + 10 z 7 a 7 -10 z 5 a 7 -10 z 3 a 7 + 20 z a 7 -15 a 7 z -1 + 4 a 7 z -3 + 10 z 6 a 6 -14 z 4 a 6 + 10 z 2 a 6 + 5 a 6 z -2 - a 6 z -4 -10 a 6 + 4 z 5 a 5 + z 4 a 4$ (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 =

-4 is the signature of L10a174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$
$r = -10$	${\mathbb Z}$
$r = -9$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{6}$	${\mathbb Z}^{5}$
$r = -7$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -4$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2$	${\mathbb Z}^{11}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$