L11a340

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Link Presentations

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

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

5

1

-4

10

7

2

5

8

9

5

-4

6

11

7

4

8

9

1

2

9

11

-2

0

6

10

4

-2

2

7

-5

-4

1

6

5

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$