L11a300

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+9t(2)^{2}t(1)^{2}-7t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-7t(2)^{2}t(1)+9t(2)t(1)-3t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$15q^{9/2}-18q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-4q^{15/2}+8q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+7z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}+2az-4za^{-1}+6za^{-3}-3za^{-5}+za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-2z^{3}a^{-9}+8z^{6}a^{-8}-8z^{4}a^{-8}+2z^{2}a^{-8}+10z^{7}a^{-7}-13z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+8z^{8}a^{-6}-6z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+4z^{9}a^{-5}+7z^{7}a^{-5}-30z^{5}a^{-5}+23z^{3}a^{-5}-7za^{-5}+a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-33z^{6}a^{-4}+21z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+7z^{9}a^{-3}-11z^{7}a^{-3}-11z^{5}a^{-3}+19z^{3}a^{-3}-6za^{-3}+a^{-3}z^{-1}+z^{10}a^{-2}+7z^{8}a^{-2}-31z^{6}a^{-2}+32z^{4}a^{-2}-9z^{2}a^{-2}+3z^{9}a^{-1}+az^{7}-7z^{7}a^{-1}-4az^{5}-2z^{5}a^{-1}+5az^{3}+10z^{3}a^{-1}-2az-3za^{-1}+3z^{8}-12z^{6}+15z^{4}-6z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

3

14

5

1

-4

12

7

3

4

10

9

6

-3

8

9

6

3

6

7

9

2

4

7

9

-2

2

4

8

4

0

2

6

-4

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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