L11n183

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_21,1,22,6 X_7,18,8,19 X_3,10,4,11 X_15,12,16,13 X_5,14,6,15 X_13,4,14,5 X_11,16,12,17 X_17,22,18,7 X_2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {-4, -1, 2, 5, -9, 6, -8, 7, -6, 9, -10, 4, 11, -2, -3, 10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-4t(2)^{2}t(1)+7t(2)t(1)-4t(1)+t(2)^{2}-3t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-a^{9}z+a^{7}z^{5}+2a^{7}z^{3}+a^{7}z-a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+3a^{5}z+3a^{5}z^{-1}-2a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-2z^{2}a^{12}-3z^{7}a^{11}+10z^{5}a^{11}-8z^{3}a^{11}+za^{11}-3z^{8}a^{10}+8z^{6}a^{10}-4z^{4}a^{10}+z^{2}a^{10}-z^{9}a^{9}-3z^{7}a^{9}+14z^{5}a^{9}-9z^{3}a^{9}+za^{9}-5z^{8}a^{8}+12z^{6}a^{8}-8z^{4}a^{8}+2z^{2}a^{8}+a^{8}-z^{9}a^{7}-2z^{7}a^{7}+7z^{5}a^{7}-6z^{3}a^{7}+2za^{7}-a^{7}z^{-1}-2z^{8}a^{6}+2z^{6}a^{6}-z^{4}a^{6}-4z^{2}a^{6}+3a^{6}-2z^{7}a^{5}+3z^{5}a^{5}-8z^{3}a^{5}+7za^{5}-3a^{5}z^{-1}-z^{6}a^{4}-3z^{2}a^{4}+3a^{4}-3z^{3}a^{3}+5za^{3}-2a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

1

-1

-6

5

1

4

-8

4

3

-1

-10

5

4

1

-12

4

0

-14

3

5

-2

-16

2

4

2

-18

1

3

-2

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$