L11a77

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,12,15,11 X_22,15,5,16 X_16,7,17,8 X_20,17,21,18 X_18,10,19,9 X_8,20,9,19 X_10,21,11,22 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -8, 7, -9, 3, -2, 11, -3, 4, -5, 6, -7, 8, -6, 9, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-13q^{3/2}+16{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}+z^{3}a^{-3}-7a^{3}z+za^{-3}-3a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+8az^{3}-5z^{3}a^{-1}+8az-4za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-5a^{4}z^{8}-11a^{2}z^{8}-6z^{8}a^{-2}-12z^{8}-5a^{5}z^{7}-8a^{3}z^{7}-z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+20a^{2}z^{6}+13z^{6}a^{-2}-z^{6}a^{-4}+29z^{6}-a^{7}z^{5}+7a^{5}z^{5}+25a^{3}z^{5}+29az^{5}+22z^{5}a^{-1}+10z^{5}a^{-3}+5a^{6}z^{4}+6a^{4}z^{4}-7a^{2}z^{4}-5z^{4}a^{-2}+2z^{4}a^{-4}-15z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-26a^{3}z^{3}-36az^{3}-23z^{3}a^{-1}-7z^{3}a^{-3}-3a^{6}z^{2}-7a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-a^{7}z+2a^{5}z+13a^{3}z+17az+9za^{-1}+2za^{-3}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

8

3

5

2

8

5

-3

0

12

8

4

-2

9

10

1

-4

7

10

-3

-6

5

9

4

-8

2

7

-5

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$