L11n451

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_20,12,21,11 X_22,13,19,14 X_18,22,9,21 X_12,17,13,18 X_15,8,16,5 X_7,14,8,15 X_16,19,17,20 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {9, -3, 5, -4}, {-11, 2, 3, -6, 4, 8, -7, -9, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v w u 3 - w u 3 -2 v w u 2 + 2 w u 2 - v x u 2 + x u 2 + v w u - w u + 2 v x u -2 x u - v x + x$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 7 z a 5 + 5 a 5 z -1 + a 5 z -3 -2 z 5 a 3 -9 z 3 a 3 -15 z a 3 -10 a 3 z -1 -3 a 3 z -3 + z 5 a + 7 z 3 a + 12 z a + 9 a z -1 + 3 a z -3 - z 3 a -1 -3 z a -1 -3 a -1 z -1 - a -1 z -3$ (db)
Kauffman polynomial	$- z 6 a 8 + 3 z 4 a 8 -3 z 2 a 8 + a 8 -3 z 7 a 7 + 10 z 5 a 7 -9 z 3 a 7 + 5 z a 7 -2 a 7 z -1 -2 z 8 a 6 + z 6 a 6 + 15 z 4 a 6 -15 z 2 a 6 + 6 a 6 -9 z 7 a 5 + 35 z 5 a 5 -43 z 3 a 5 + 31 z a 5 -11 a 5 z -1 + a 5 z -3 -2 z 8 a 4 + z 6 a 4 + 19 z 4 a 4 -33 z 2 a 4 -3 a 4 z -2 + 18 a 4 -7 z 7 a 3 + 39 z 5 a 3 -73 z 3 a 3 + 54 z a 3 -18 a 3 z -1 + 3 a 3 z -3 - z 8 a 2 + 5 z 6 a 2 + 2 z 4 a 2 -27 z 2 a 2 -6 a 2 z -2 + 21 a 2 -2 z 7 a + 20 z 5 a -49 z 3 a + 38 z a -14 a z -1 + 3 a z -3 - z 8 + 6 z 6 -5 z 4 -6 z 2 -3 z -2 + 9- z 7 a -1 + 6 z 5 a -1 -10 z 3 a -1 + 10 z a -1 -5 a -1 z -1 + a -1 z -3$ (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 =

-1 is the signature of L11n451. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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