L11n34

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,10,15,9 X₈₄₉₃ X_5,13,6,12 X_13,5,14,22 X_18,16,19,15 X_16,11,17,12 X_10,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, 4, -10, 9, 6, -7, -4, 8, -9, 10, -8, 11, -2, 3, 7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {4}{q^{7/2}}}-11q^{5/2}+{\frac {7}{q^{5/2}}}+14q^{3/2}-{\frac {12}{q^{3/2}}}-15{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+4za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}+2az-4za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$6z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+a^{4}z^{6}+3z^{6}a^{-4}-2a^{4}z^{4}+3z^{4}a^{-4}+a^{4}z^{2}-3z^{2}a^{-4}+a^{-4}+4a^{3}z^{7}+8z^{7}a^{-3}-11a^{3}z^{5}-16z^{5}a^{-3}+9a^{3}z^{3}+24z^{3}a^{-3}-a^{3}z-15za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+5a^{2}z^{8}+7z^{8}a^{-2}-10a^{2}z^{6}-10z^{6}a^{-2}+2a^{2}z^{4}+8z^{4}a^{-2}+a^{2}z^{2}-6z^{2}a^{-2}+a^{2}+3a^{-2}+2az^{9}+2z^{9}a^{-1}+9az^{7}+13z^{7}a^{-1}-35az^{5}-40z^{5}a^{-1}+27az^{3}+36z^{3}a^{-1}-5az-14za^{-1}-az^{-1}+a^{-1}z^{-1}+12z^{8}-24z^{6}+9z^{4}-3z^{2}+2$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over

r

).

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

3

8

4

-4

6

7

3

4

7

4

-3

2

8

7

1

0

8

9

1

-2

4

6

-2

-4

3

8

5

-6

1

4

-3

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

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