L11a359

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Link Presentations

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

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

Polynomial invariants

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

14

1

12

2

-2

10

4

1

3

8

6

2

-4

6

8

4

8

7

-1

2

8

7

1

0

6

9

3

-2

4

7

-3

-4

2

6

4

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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