L11a285

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}+2uv^{3}-3uv^{2}+2uv+v^{2}-v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-11q^{3/2}+14{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {16}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-18az^{5}+5z^{5}a^{-1}+8a^{3}z^{3}-19az^{3}+8z^{3}a^{-1}+3a^{3}z-6az+3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-9az^{9}-4z^{9}a^{-1}-6a^{4}z^{8}-2a^{2}z^{8}-4z^{8}a^{-2}-5a^{5}z^{7}+11a^{3}z^{7}+28az^{7}+9z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+13a^{4}z^{6}+11a^{2}z^{6}+9z^{6}a^{-2}-z^{6}a^{-4}+5z^{6}-a^{7}z^{5}+10a^{5}z^{5}-15a^{3}z^{5}-44az^{5}-9z^{5}a^{-1}+9z^{5}a^{-3}+6a^{6}z^{4}-12a^{4}z^{4}-17a^{2}z^{4}-3z^{4}a^{-2}+3z^{4}a^{-4}-5z^{4}+2a^{7}z^{3}-6a^{5}z^{3}+8a^{3}z^{3}+32az^{3}+9z^{3}a^{-1}-7z^{3}a^{-3}-2a^{6}z^{2}+2a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+4z^{2}+a^{5}z-2a^{3}z-6az-2za^{-1}+za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

7

2

5

2

7

4

-3

0

10

7

3

-2

8

9

1

-4

7

8

-1

-6

4

8

4

-8

2

7

-5

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$