L11a518

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_22,9,13,10 X_20,11,21,12 X_12,19,7,20 X_10,21,11,22 X_18,14,19,13 X_16,6,17,5 X₂₇₃₈ X_4,16,5,15 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, {7, -2, 10, -8, 11, -7, 5, -4, 6, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-t(1)t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+2t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-2t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)t(2)^{2}+2t(2)^{2}+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}+q^{-5}-5q^{4}-q^{-4}+8q^{3}+4q^{-3}-9q^{2}-5q^{-2}+10q+8q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}-2a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}+a^{4}z^{2}-7a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}+3a^{4}-6a^{2}+a^{-2}+2+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+3z^{4}a^{-6}-z^{2}a^{-6}+5z^{5}a^{-5}-3z^{3}a^{-5}+a^{4}z^{8}-7a^{4}z^{6}+7z^{6}a^{-4}+17a^{4}z^{4}-10z^{4}a^{-4}-17a^{4}z^{2}+4z^{2}a^{-4}-a^{4}z^{-2}+7a^{4}+a^{3}z^{9}-4a^{3}z^{7}+7z^{7}a^{-3}+a^{3}z^{5}-13z^{5}a^{-3}+10a^{3}z^{3}+5z^{3}a^{-3}-9a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}-3a^{2}z^{8}+5z^{8}a^{-2}-3a^{2}z^{6}-8z^{6}a^{-2}+17a^{2}z^{4}-5z^{4}a^{-2}-20a^{2}z^{2}+7z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-2a^{-2}+4az^{9}+3z^{9}a^{-1}-16az^{7}-5z^{7}a^{-1}+15az^{5}-4z^{5}a^{-1}+4az^{3}+3z^{3}a^{-1}-9az+2az^{-1}+z^{10}+z^{8}-11z^{6}+8z^{4}-z^{2}-z^{-2}+3$ (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

13

1

-1

11

2

9

3

1

-2

7

5

2

3

5

4

-1

3

5

4

1

6

7

1

-1

2

3

-1

-3

3

6

3

-5

1

2

-1

-7

3

-9

1

0

-11

1

Integral Khovanov Homology

(db, data source)

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