L11n171

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Planar diagram presentation	X₈₁₉₂ X_11,21,12,20 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_13,19,14,18 X_22,16,7,15 X_19,13,20,12 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+6t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$2q^{11/2}-5q^{9/2}+9q^{7/2}-12q^{5/2}+13q^{3/2}-14{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+z^{3}a^{-3}-2za^{-3}-2a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-5z^{3}a^{-1}+az-za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-9z^{8}a^{-2}-3z^{8}a^{-4}-6z^{8}-7az^{7}-6z^{7}a^{-1}-z^{7}a^{-5}-4a^{2}z^{6}+20z^{6}a^{-2}+6z^{6}a^{-4}+10z^{6}-a^{3}z^{5}+14az^{5}+18z^{5}a^{-1}-3z^{5}a^{-5}+6a^{2}z^{4}-22z^{4}a^{-2}-14z^{4}a^{-4}-3z^{4}a^{-6}-5z^{4}+a^{3}z^{3}-7az^{3}-12z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+12z^{2}a^{-2}+15z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}+2az-3za^{-3}-za^{-5}-3a^{-2}-5a^{-4}-2a^{-6}+1-az^{-1}+2a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

2

-2

10

3

8

6

2

-4

6

3

4

7

6

-1

2

7

6

1

0

4

8

4

-2

4

6

-2

-4

1

5

4

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

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