L11a317

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_20,11,21,12 X_6,9,7,10 X_16,7,17,8 X_8,15,1,16 X_22,17,9,18 X_12,4,13,3 X_18,6,19,5 X_4,14,5,13 X_14,21,15,22 X_2,20,3,19
Gauss code	{1, -11, 7, -9, 8, -3, 4, -5}, {3, -1, 2, -7, 9, -10, 5, -4, 6, -8, 11, -2, 10, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{3}-2u^{2}v^{2}+u^{2}v+uv^{4}-3uv^{3}+6uv^{2}-3uv+u+v^{3}-2v^{2}+v\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {17}{q^{9/2}}}-q^{7/2}+{\frac {24}{q^{7/2}}}+5q^{5/2}-{\frac {29}{q^{5/2}}}-12q^{3/2}+{\frac {28}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {10}{q^{11/2}}}+19{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+5a^{3}z^{3}-z^{3}a^{-1}-2a^{5}z+5a^{3}z-3az-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-4a^{4}z^{10}-4a^{2}z^{10}-9a^{5}z^{9}-21a^{3}z^{9}-12az^{9}-8a^{6}z^{8}-11a^{4}z^{8}-19a^{2}z^{8}-16z^{8}-4a^{7}z^{7}+16a^{5}z^{7}+40a^{3}z^{7}+8az^{7}-12z^{7}a^{-1}-a^{8}z^{6}+17a^{6}z^{6}+38a^{4}z^{6}+48a^{2}z^{6}-5z^{6}a^{-2}+23z^{6}+8a^{7}z^{5}-11a^{5}z^{5}-26a^{3}z^{5}+9az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-13a^{6}z^{4}-32a^{4}z^{4}-29a^{2}z^{4}+3z^{4}a^{-2}-9z^{4}-4a^{7}z^{3}+7a^{5}z^{3}+15a^{3}z^{3}-az^{3}-5z^{3}a^{-1}-a^{8}z^{2}+5a^{6}z^{2}+12a^{4}z^{2}+8a^{2}z^{2}+2z^{2}-4a^{5}z-9a^{3}z-5az-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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

8

1

6

4

-4

4

8

1

7

2

11

4

-7

0

15

8

7

-2

15

13

-2

-4

14

13

1

-6

10

15

5

-8

7

14

-7

-10

3

10

7

-12

1

7

-6

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$