L11n258

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_19,22,20,9 X_15,20,16,21 X_21,16,22,17 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 -3 u 3 -2 v u 2 + 4 v w u 2 -2 w u 2 + 5 u 2 + 2 v u -5 v w u + 2 w u -4 u - v + 3 v w - w + 1$ (db)
Jones polynomial	$q 3 -2 q 2 + 7 q -9 + 13 q -1 -12 q -2 + 13 q -3 -10 q -4 + 6 q -5 -3 q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 6 z -2 -2 a 6 - z 4 a 4 + 2 z 2 a 4 + 3 a 4 z -2 + 6 a 4 + z 6 a 2 + 2 z 4 a 2 -2 a 2 z -2 -3 a 2 -2 z 4 -4 z 2 - z -2 -3 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 5 a 4 z 8 + 8 a 2 z 8 + 3 z 8 + 7 a 5 z 7 + 14 a 3 z 7 + 9 a z 7 + 2 z 7 a -1 + 3 a 6 z 6 -4 a 4 z 6 -13 a 2 z 6 + z 6 a -2 -5 z 6 -17 a 5 z 5 -43 a 3 z 5 -30 a z 5 -4 z 5 a -1 -2 a 4 z 4 + a 2 z 4 -4 z 4 a -2 - z 4 + 6 a 7 z 3 + 32 a 5 z 3 + 51 a 3 z 3 + 25 a z 3 + a 6 z 2 + a 4 z 2 -2 a 2 z 2 + 6 z 2 a -2 + 4 z 2 -7 a 7 z -27 a 5 z -34 a 3 z -10 a z + 4 z a -1 + a 6 + 5 a 4 + 4 a 2 -4 a -2 -3 + 2 a 7 z -1 + 8 a 5 z -1 + 10 a 3 z -1 + 2 a z -1 -2 a -1 z -1 - a 6 z -2 -3 a 4 z -2 -2 a 2 z -2 + a -2 z -2 + z -2$ (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 L11n258. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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