L11a415

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,10,19,9 X_16,8,17,7 X_22,16,11,15 X_20,14,21,13 X_14,22,15,21 X_10,18,5,17 X_8,20,9,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 6, -7, 5, -4, 8, -3, 9, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 4 - v w u 4 + w u 4 - u 4 + 2 v 2 u 3 -4 v u 3 - v 2 w u 3 + 3 v w u 3 -2 w u 3 + u 3 -2 v 2 u 2 + 4 v u 2 + v 2 w u 2 -4 v w u 2 + 2 w u 2 - u 2 + 2 v 2 u -3 v u - v 2 w u + 4 v w u -2 w u + u - v 2 + v + v 2 w -2 v w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -7 q 6 + 11 q 5 -14 q 4 + 17 q 3 -15 q 2 + 14 q -9 + 6 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 -4 z 2 a -2 + 6 z 2 a -4 -2 z 2 a -6 -4 z 2 + 2 a 2 -8 a -2 + 8 a -4 -2 a -6 -5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 5 z 8 a -2 + 9 z 8 a -4 + 7 z 8 a -6 + 3 z 8 + 2 a z 7 + 2 z 7 a -1 -8 z 7 a -3 -2 z 7 a -5 + 6 z 7 a -7 + a 2 z 6 -17 z 6 a -2 -30 z 6 a -4 -18 z 6 a -6 + 3 z 6 a -8 -7 z 6 -5 a z 5 -16 z 5 a -1 -9 z 5 a -3 -13 z 5 a -5 -14 z 5 a -7 + z 5 a -9 -4 a 2 z 4 + 27 z 4 a -2 + 44 z 4 a -4 + 22 z 4 a -6 -5 z 4 a -8 + 6 z 4 + 2 a z 3 + 21 z 3 a -1 + 35 z 3 a -3 + 30 z 3 a -5 + 12 z 3 a -7 -2 z 3 a -9 + 5 a 2 z 2 -33 z 2 a -2 -33 z 2 a -4 -12 z 2 a -6 -7 z 2 + a z -16 z a -1 -33 z a -3 -21 z a -5 -5 z a -7 -2 a 2 + 20 a -2 + 17 a -4 + 4 a -6 + 6 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

2 is the signature of L11a415. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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