L11n241

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_18,14,19,13 X_22,20,11,19 X_20,15,21,16 X_14,21,15,22 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-t(2)t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$7q^{9/2}-9q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}+2q^{13/2}-5q^{11/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-4z^{3}a^{-5}-6za^{-5}-4a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+12za^{-3}+6a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-10za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{2}a^{-8}-a^{-8}+z^{5}a^{-7}+5z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+5z^{6}a^{-6}-9z^{4}a^{-6}+9z^{2}a^{-6}-3a^{-6}+8z^{7}a^{-5}-25z^{5}a^{-5}+31z^{3}a^{-5}-18za^{-5}+4a^{-5}z^{-1}+5z^{8}a^{-4}-10z^{6}a^{-4}-2z^{4}a^{-4}+8z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+10z^{7}a^{-3}-47z^{5}a^{-3}+58z^{3}a^{-3}-31za^{-3}+6a^{-3}z^{-1}+7z^{8}a^{-2}-23z^{6}a^{-2}+16z^{4}a^{-2}-a^{-2}+z^{9}a^{-1}+az^{7}+3z^{7}a^{-1}-5az^{5}-26z^{5}a^{-1}+9az^{3}+41z^{3}a^{-1}-7az-24za^{-1}+2az^{-1}+5a^{-1}z^{-1}+2z^{8}-8z^{6}+9z^{4}-2z^{2}-1$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

2

-2

12

3

10

4

2

-2

8

5

3

2

6

4

0

4

5

0

2

4

6

2

0

1

3

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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