L11n441

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L11n440

L11n442

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

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[edit] Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, -9, 10, 11, -3, -8, 9, -10, 8}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 t(1) t(4)^3-t(1) t(2) t(4)^3+t(2) t(4)^3-t(1) t(3) t(4)^3-t(2) t(3) t(4)^3+t(3) t(4)^3-t(4)^3-2 t(1) t(4)^2+t(1) t(2) t(4)^2-t(2) t(4)^2+t(1) t(3) t(4)^2+2 t(2) t(3) t(4)^2-t(3) t(4)^2+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)-t(1) t(3) t(4)-2 t(2) t(3) t(4)+t(3) t(4)-t(1)+t(1) t(2)-t(2)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}}$ (db)
Jones polynomial	$-\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{11/2}}+3 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + 2 a 9 z -3 -4 z a 7 -11 a 7 z -1 -7 a 7 z -3 + 6 z 3 a 5 + 20 z a 5 + 23 a 5 z -1 + 9 a 5 z -3 -3 z 5 a 3 -13 z 3 a 3 -22 z a 3 -17 a 3 z -1 -5 a 3 z -3 + 3 z 3 a + 6 z a + 4 a z -1 + a z -3$ (db)
Kauffman polynomial	$a 9 z 7 -6 a 9 z 5 + 14 a 9 z 3 -2 a 9 z -3 -16 a 9 z + 9 a 9 z -1 + a 8 z 8 - a 8 z 6 -10 a 8 z 4 + 26 a 8 z 2 + 7 a 8 z -2 -23 a 8 + a 7 z 9 + 2 a 7 z 7 -21 a 7 z 5 + 41 a 7 z 3 -7 a 7 z -3 -39 a 7 z + 23 a 7 z -1 + 6 a 6 z 8 -10 a 6 z 6 -29 a 6 z 4 + 73 a 6 z 2 + 19 a 6 z -2 -60 a 6 + a 5 z 9 + 12 a 5 z 7 -47 a 5 z 5 + 51 a 5 z 3 -9 a 5 z -3 -37 a 5 z + 24 a 5 z -1 + 5 a 4 z 8 + a 4 z 6 -44 a 4 z 4 + 75 a 4 z 2 + 18 a 4 z -2 -58 a 4 + 11 a 3 z 7 -29 a 3 z 5 + 27 a 3 z 3 -5 a 3 z -3 -16 a 3 z + 12 a 3 z -1 + 10 a 2 z 6 -25 a 2 z 4 + 34 a 2 z 2 + 7 a 2 z -2 -24 a 2 + 3 a z 5 + 3 a z 3 - a z -3 -2 a z + 2 a z -1 + 6 z 2 + z -2 -4$ (db)

[edit] Khovanov Homology

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 L11n441. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11n441/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -8$		${\mathbb Z}$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$		${\mathbb Z}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$		${\mathbb Z}^{11}\oplus{\mathbb Z}_2$	${\mathbb Z}^{7}$
$r = -3$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$		${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$		${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 1$		${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$