L11n444

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_13,20,14,21 X_19,15,20,22 X_21,12,22,13 X_17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -3, 10, -8, 11}, {7, -6, -11, 3, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(2)t(3)^{2}+t(1)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-2t(1)t(4)t(3)-2t(2)t(4)t(3)-t(1)t(4)^{2}-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {6}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+a^{9}z^{-3}-4za^{7}-7a^{7}z^{-1}-3a^{7}z^{-3}+5z^{3}a^{5}+14za^{5}+12a^{5}z^{-1}+3a^{5}z^{-3}-2z^{5}a^{3}-8z^{3}a^{3}-13za^{3}-7a^{3}z^{-1}-a^{3}z^{-3}+2z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-10z^{3}a^{9}+10za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-2z^{8}a^{8}+7z^{6}a^{8}-4z^{4}a^{8}-7z^{2}a^{8}-3a^{8}z^{-2}+9a^{8}-z^{9}a^{7}-4z^{7}a^{7}+31z^{5}a^{7}-50z^{3}a^{7}+35za^{7}-14a^{7}z^{-1}+3a^{7}z^{-3}-7z^{8}a^{6}+21z^{6}a^{6}-4z^{4}a^{6}-24z^{2}a^{6}-6a^{6}z^{-2}+21a^{6}-z^{9}a^{5}-11z^{7}a^{5}+54z^{5}a^{5}-74z^{3}a^{5}+49za^{5}-18a^{5}z^{-1}+3a^{5}z^{-3}-5z^{8}a^{4}+9z^{6}a^{4}+11z^{4}a^{4}-28z^{2}a^{4}-3a^{4}z^{-2}+18a^{4}-8z^{7}a^{3}+27z^{5}a^{3}-39z^{3}a^{3}+30za^{3}-11a^{3}z^{-1}+a^{3}z^{-3}-5z^{6}a^{2}+11z^{4}a^{2}-14z^{2}a^{2}+6a^{2}-z^{5}a-5z^{3}a+6za-2az^{-1}-3z^{2}+1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

-2

0

3

-2

4

3

-1

-4

6

3

1

4

-6

5

7

2

-8

5

4

1

2

-10

4

8

4

-12

2

0

-14

1

5

4

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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