L11a495

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-7q^{-5}+q^{4}+12q^{-4}-4q^{3}-16q^{-3}+10q^{2}+21q^{-2}-14q-20q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+2a^{4}z^{4}+3a^{4}z^{2}+a^{4}-a^{2}z^{6}-a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-z^{6}-2z^{4}-4z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+3a^{6}z^{8}-11a^{6}z^{6}+13a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+4a^{5}z^{9}-11a^{5}z^{7}+5a^{5}z^{5}+5a^{5}z^{3}-3a^{5}z+2a^{4}z^{10}+5a^{4}z^{8}-33a^{4}z^{6}+41a^{4}z^{4}+z^{4}a^{-4}-17a^{4}z^{2}+2a^{4}+12a^{3}z^{9}-31a^{3}z^{7}+21a^{3}z^{5}+4z^{5}a^{-3}-6a^{3}z^{3}+3a^{3}z+2a^{2}z^{10}+15a^{2}z^{8}-48a^{2}z^{6}+10z^{6}a^{-2}+34a^{2}z^{4}-10z^{4}a^{-2}-a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-6a^{2}-4a^{-2}+8az^{9}-5az^{7}+14z^{7}a^{-1}-10az^{5}-18z^{5}a^{-1}-2az^{3}+4z^{3}a^{-1}+7az+3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+13z^{8}-16z^{6}-5z^{4}+15z^{2}+2z^{-2}-9$ (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

9

1

7

3

-3

5

7

1

6

3

7

3

-4

1

12

7

5

-1

12

11

-1

-3

9

8

1

-5

7

12

5

-7

5

9

-4

-9

2

7

5

-11

1

5

-4

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$