L11n317

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,7,17,8 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_8,15,9,16 X_21,10,22,5 X_14,18,15,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, -4, 8}, {-5, -2, 11, -9, 7, -3, 9, 5, -6, 4, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -2 v u 2 - v 2 w u 2 + 3 v w u 2 - w u 2 + u 2 -3 v 2 u + 5 v u + 2 v 2 w u -5 v w u + 3 w u -2 u + v 2 -3 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$- q 2 + 4 q -7 + 11 q -1 -12 q -2 + 14 q -3 -11 q -4 + 9 q -5 -5 q -6 + 2 q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2 z 2 a 6 + a 6 z -2 + 2 a 6 -3 z 4 a 4 -7 z 2 a 4 -2 a 4 z -2 -6 a 4 + z 6 a 2 + 3 z 4 a 2 + 5 z 2 a 2 + a 2 z -2 + 4 a 2 - z 4 - z 2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 6 a 4 z 8 + 4 a 2 z 8 + a 7 z 7 + 5 a 5 z 7 + 10 a 3 z 7 + 6 a z 7 + a 6 z 6 -2 a 4 z 6 + a 2 z 6 + 4 z 6 + 3 a 7 z 5 -5 a 5 z 5 -19 a 3 z 5 -10 a z 5 + z 5 a -1 + 3 a 8 z 4 -3 a 6 z 4 -13 a 4 z 4 -14 a 2 z 4 -7 z 4 -4 a 7 z 3 -4 a 5 z 3 + 4 a 3 z 3 + 3 a z 3 - z 3 a -1 -4 a 8 z 2 + 4 a 6 z 2 + 16 a 4 z 2 + 11 a 2 z 2 + 3 z 2 + 6 a 5 z + 6 a 3 z + a 8 -3 a 6 -8 a 4 -5 a 2 -2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

-2 is the signature of L11n317. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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