L11n458

(Knotscape image)

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

Link L11n458.

A graph, L11n458.

A part of a knot and a part of a graph.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 -2 v w x u 2 + w x u 2 -2 v 2 u + 3 v u + v 2 w u -3 v w u + 2 w u + 2 v 2 x u -3 v x u + 3 v w x u -2 w x u + x u + v 2 -2 v - v 2 x + 2 v x - v w x + w x - x$ (db)
Jones polynomial	$q 21 / 2 -4 q 19 / 2 + 7 q 17 / 2 -11 q 15 / 2 + 12 q 13 / 2 -15 q 11 / 2 + 11 q 9 / 2 -11 q 7 / 2 + 5 q 5 / 2 -3 q 3 / 2$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2 z 5 a -5 - z 5 a -7 + 3 z 3 a -3 -6 z 3 a -5 + z 3 a -9 + 6 z a -3 -10 z a -5 + 4 z a -7 + 4 a -3 z -1 -9 a -5 z -1 + 6 a -7 z -1 - a -9 z -1 + a -3 z -3 -3 a -5 z -3 + 3 a -7 z -3 - a -9 z -3$ (db)
Kauffman polynomial	$-2 z 9 a -7 -2 z 9 a -9 -6 z 8 a -6 -11 z 8 a -8 -5 z 8 a -10 -7 z 7 a -5 -10 z 7 a -7 -7 z 7 a -9 -4 z 7 a -11 -3 z 6 a -4 + 10 z 6 a -6 + 25 z 6 a -8 + 11 z 6 a -10 - z 6 a -12 + 19 z 5 a -5 + 41 z 5 a -7 + 33 z 5 a -9 + 11 z 5 a -11 + 3 z 4 a -4 -2 z 4 a -6 -9 z 4 a -8 -2 z 4 a -10 + 2 z 4 a -12 -6 z 3 a -3 -34 z 3 a -5 -52 z 3 a -7 -32 z 3 a -9 -8 z 3 a -11 -9 z 2 a -4 -18 z 2 a -6 -10 z 2 a -8 -2 z 2 a -10 - z 2 a -12 + 10 z a -3 + 27 z a -5 + 31 z a -7 + 16 z a -9 + 2 z a -11 + 10 a -4 + 19 a -6 + 10 a -8 -5 a -3 z -1 -12 a -5 z -1 -12 a -7 z -1 -5 a -9 z -1 -3 a -4 z -2 -6 a -6 z -2 -3 a -8 z -2 + a -3 z -3 + 3 a -5 z -3 + 3 a -7 z -3 + a -9 z -3$ (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 =

3 is the signature of L11n458. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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