L11n25

(Knotscape image)

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Visit L11n25's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_20,9,21,10 X₈₄₉₃ X_18,22,19,21 X_11,14,12,15 X_5,13,6,12 X_13,5,14,22 X_10,19,11,20 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, 4, -10, -7, 8, -9, 7, 11, -2, 3, -6, 10, -4, 6, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 7 v u 2 -7 u 2 -7 v u + 7 u + 2 v -2$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 + z 3 a 3 -2 z a 3 -2 a 3 z -1 - z 5 a + 2 z a + a z -1 - z 5 a -1 - z 3 a -1 + a -1 z -1 + z 3 a -3 - a -3 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -5 z 8 a -2 -10 z 8 -4 a 3 z 7 -4 a z 7 -4 z 7 a -1 -4 z 7 a -3 - a 4 z 6 + 12 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 28 z 6 + 7 a 3 z 5 + 19 a z 5 + 24 z 5 a -1 + 12 z 5 a -3 -4 a 4 z 4 -19 a 2 z 4 -10 z 4 a -2 + 2 z 4 a -4 -27 z 4 -3 a 5 z 3 -14 a 3 z 3 -22 a z 3 -19 z 3 a -1 -8 z 3 a -3 + 3 a 4 z 2 + 13 a 2 z 2 + 4 z 2 a -2 + 14 z 2 + 3 a 5 z + 11 a 3 z + 10 a z + z a -1 - z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 L11n25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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