L11a492

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_22,16,19,15 X_20,9,21,10 X_8,19,9,20 X_4,17,1,18 X_12,6,13,5 X_10,4,11,3 X_18,12,5,11 X_16,22,17,21 X_2,14,3,13
Gauss code	{1, -11, 8, -6}, {5, -4, 10, -3}, {7, -1, 2, -5, 4, -8, 9, -7, 11, -2, 3, -10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-22q+25-19q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-7z^{4}a^{-2}-3a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-a^{2}-3a^{-2}+z^{8}+5z^{6}+10z^{4}+9z^{2}-2z^{-2}+3$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+8az^{9}+14z^{9}a^{-1}+6z^{9}a^{-3}+11a^{2}z^{8}+16z^{8}a^{-2}+7z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}-8az^{7}-24z^{7}a^{-1}-4z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-23a^{2}z^{6}-55z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-66z^{6}+a^{5}z^{5}-12a^{3}z^{5}-5az^{5}-17z^{5}a^{-3}-9z^{5}a^{-5}-5a^{4}z^{4}+24a^{2}z^{4}+60z^{4}a^{-2}+9z^{4}a^{-4}-2z^{4}a^{-6}+78z^{4}-a^{5}z^{3}+4a^{3}z^{3}+13az^{3}+21z^{3}a^{-1}+19z^{3}a^{-3}+6z^{3}a^{-5}-13a^{2}z^{2}-30z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-36z^{2}-a^{3}z-5az-9za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+6a^{-2}+2a^{-4}+7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

6

1

5

7

9

3

-6

5

11

6

5

3

11

9

-2

1

14

11

3

-1

9

15

6

-3

6

10

-4

-5

3

9

6

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

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