L11n315

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_15,9,16,8 X_21,10,22,5 X_17,14,18,15 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 $u$ , $v$ , $w$ , ...)	$v u 2 + v w u 2 - w u 2 - u 2 - v 2 u + v u - v 2 w u - v w u + w u + u + v 2 - v + v 2 w - v w$ (db)
Jones polynomial	$q 3 - q 2 + 2 q + q -1 + q -2 - q -3 + 2 q -4 -2 q -5 + 2 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + z 4 a 4 + 3 z 2 a 4 + 2 a 4 + a 2 z -2 + a 2 - z 4 -4 z 2 -2 z -2 -4 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 3 a 4 z 8 + a 2 z 8 + a 7 z 7 -4 a 5 z 7 -6 a 3 z 7 + z 7 a -1 -11 a 6 z 6 -19 a 4 z 6 -7 a 2 z 6 + z 6 a -2 + 2 z 6 -5 a 7 z 5 - a 5 z 5 + 8 a 3 z 5 -4 z 5 a -1 + 16 a 6 z 4 + 33 a 4 z 4 + 11 a 2 z 4 -5 z 4 a -2 -11 z 4 + 6 a 7 z 3 + 9 a 5 z 3 -3 a 3 z 3 -5 a z 3 + z 3 a -1 -8 a 6 z 2 -20 a 4 z 2 -5 a 2 z 2 + 7 z 2 a -2 + 14 z 2 -2 a 7 z -4 a 5 z + 6 a z + 4 z a -1 + 2 a 6 + 4 a 4 -2 a 2 -4 a -2 -7-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L11n315. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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