L11a262

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-4t(2)t(1)+2t(1)-t(2)^{2}+2t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+12q^{7/2}-{\frac {3}{q^{7/2}}}-16q^{5/2}+{\frac {6}{q^{5/2}}}+18q^{3/2}-{\frac {11}{q^{3/2}}}-q^{13/2}+4q^{11/2}-18{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}-a^{3}z^{3}-a^{3}z-za^{-3}+az^{5}+2z^{5}a^{-1}+az^{3}+3z^{3}a^{-1}+az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+z^{2}a^{-6}+7z^{7}a^{-5}-12z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-10z^{6}a^{-4}-3a^{4}z^{4}+4z^{4}a^{-4}+2a^{4}z^{2}-z^{2}a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}+2z^{7}a^{-3}-9a^{3}z^{5}-13z^{5}a^{-3}+8a^{3}z^{3}+10z^{3}a^{-3}-3a^{3}z-2za^{-3}+z^{10}a^{-2}+4a^{2}z^{8}+11z^{8}a^{-2}-9a^{2}z^{6}-27z^{6}a^{-2}+4a^{2}z^{4}+23z^{4}a^{-2}-5z^{2}a^{-2}+3az^{9}+7z^{9}a^{-1}-az^{7}-9z^{7}a^{-1}-11az^{5}-2z^{5}a^{-1}+13az^{3}+9z^{3}a^{-1}-6az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+8z^{8}-23z^{6}+20z^{4}-5z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

7

3

-4

6

9

5

4

9

7

-2

2

9

0

7

11

4

-2

4

7

-3

-4

2

7

5

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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