L11a526

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(-t(2)^{2}t(1)^{2}+t(2)t(3)^{2}t(1)^{2}-t(3)^{2}t(1)^{2}+2t(2)t(1)^{2}-2t(2)t(3)t(1)^{2}+t(3)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)+t(2)^{2}t(3)^{2}t(1)-3t(2)t(3)^{2}t(1)+2t(3)^{2}t(1)-3t(2)t(1)-2t(2)^{2}t(3)t(1)+6t(2)t(3)t(1)-2t(3)t(1)+t(1)-t(2)^{2}-t(2)^{2}t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}+t(2)+t(2)^{2}t(3)-2t(2)t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+10q-16+23q^{-1}-25q^{-2}+27q^{-3}-22q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-a^{6}z^{2}-2a^{6}+a^{4}z^{6}+2a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+6a^{4}+a^{2}z^{6}-a^{2}z^{4}-6a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-7a^{2}-2z^{4}+z^{-2}+3$ (db)
Kauffman polynomial	$a^{9}z^{5}-a^{9}z^{3}+4a^{8}z^{6}-4a^{8}z^{4}+9a^{7}z^{7}-13a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+13a^{6}z^{8}-25a^{6}z^{6}+24a^{6}z^{4}-13a^{6}z^{2}+4a^{6}+10a^{5}z^{9}-10a^{5}z^{7}-5a^{5}z^{5}+13a^{5}z^{3}-6a^{5}z+3a^{4}z^{10}+20a^{4}z^{8}-64a^{4}z^{6}+73a^{4}z^{4}-42a^{4}z^{2}-a^{4}z^{-2}+13a^{4}+18a^{3}z^{9}-35a^{3}z^{7}+18a^{3}z^{5}+3a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+3a^{2}z^{10}+15a^{2}z^{8}-54a^{2}z^{6}+z^{6}a^{-2}+62a^{2}z^{4}-2z^{4}a^{-2}-40a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+8az^{9}-12az^{7}+4z^{7}a^{-1}+az^{5}-8z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-3az+2az^{-1}+8z^{8}-18z^{6}+15z^{4}-10z^{2}-z^{-2}+5$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

7

1

6

1

9

3

-6

-1

14

7

-3

13

11

-2

-5

14

12

2

-7

10

15

5

-9

7

12

-5

-11

3

10

7

-13

1

7

-6

-15

3

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$