L11n58

(Knotscape image)

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

Visit L11n58's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_18,11,19,12 X_22,19,5,20 X_20,15,21,16 X_16,21,17,22 X_12,17,13,18 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 4, -8, -9, 3, 6, -7, 8, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -3 v u 4 + 3 u 4 + 6 v u 3 -6 u 3 -6 v u 2 + 6 u 2 + 3 v u -3 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{13}{q^{13/2}}+\frac{13}{q^{15/2}}-\frac{12}{q^{17/2}}+\frac{9}{q^{19/2}}-\frac{5}{q^{21/2}}+\frac{2}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-2 z a 11 -2 a 11 z -1 + 5 z 3 a 9 + 11 z a 9 + 5 a 9 z -1 -3 z 5 a 7 -10 z 3 a 7 -10 z a 7 -3 a 7 z -1 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$-3 z 4 a 14 + 4 z 2 a 14 - a 14 - z 7 a 13 -3 z 5 a 13 + 4 z 3 a 13 - z a 13 -2 z 8 a 12 - z 6 a 12 + z 4 a 12 + z 2 a 12 - z 9 a 11 -6 z 7 a 11 + 10 z 5 a 11 -6 z 3 a 11 + 4 z a 11 -2 a 11 z -1 -6 z 8 a 10 + z 6 a 10 + 16 z 4 a 10 -17 z 2 a 10 + 5 a 10 - z 9 a 9 -11 z 7 a 9 + 27 z 5 a 9 -26 z 3 a 9 + 17 z a 9 -5 a 9 z -1 -4 z 8 a 8 - z 6 a 8 + 16 z 4 a 8 -15 z 2 a 8 + 5 a 8 -6 z 7 a 7 + 13 z 5 a 7 -14 z 3 a 7 + 11 z a 7 -3 a 7 z -1 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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