L11a449

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-4t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-5t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-13q+17-15q^{-1}+14q^{-2}-10q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}+3a^{4}z^{2}-14a^{2}z^{2}-5z^{2}a^{-2}+15z^{2}+3a^{4}-9a^{2}-2a^{-2}+8+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-2z^{3}a^{-5}-a^{5}z+4a^{4}z^{8}-8a^{4}z^{6}+3z^{6}a^{-4}+a^{4}z^{4}-6z^{4}a^{-4}-2a^{4}z^{2}+z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+3a^{3}z^{9}-a^{3}z^{7}+5z^{7}a^{-3}-13a^{3}z^{5}-11z^{5}a^{-3}+14a^{3}z^{3}+7z^{3}a^{-3}-8a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+6z^{8}a^{-2}-30a^{2}z^{6}-16z^{6}a^{-2}+39a^{2}z^{4}+21z^{4}a^{-2}-28a^{2}z^{2}-13z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}+3a^{-2}+7az^{9}+4z^{9}a^{-1}-14az^{7}-5z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+16az^{3}+16z^{3}a^{-1}-12az-7za^{-1}+2az^{-1}+z^{10}+10z^{8}-40z^{6}+62z^{4}-37z^{2}-z^{-2}+11$ (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

11

1

-1

9

2

7

4

1

-3

5

6

2

4

3

7

4

-3

1

10

6

4

-1

8

10

2

-3

6

7

-1

-5

4

8

4

-7

3

6

-3

-9

1

5

4

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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