L11a321

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(1)t(2)-t(2)+2)(2t(2)t(1)-t(1)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+3q^{5/2}-{\frac {21}{q^{5/2}}}-7q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+12{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z+a^{3}z^{7}+3a^{3}z^{5}+2a^{3}z^{3}-a^{3}z+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-3z^{3}a^{-1}+5az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+5a^{7}z^{3}-a^{7}z+7a^{6}z^{8}-16a^{6}z^{6}+9a^{6}z^{4}-3a^{6}z^{2}+6a^{5}z^{9}-8a^{5}z^{7}-5a^{5}z^{5}+5a^{5}z^{3}-a^{5}z+2a^{4}z^{10}+10a^{4}z^{8}-32a^{4}z^{6}+24a^{4}z^{4}-5a^{4}z^{2}+11a^{3}z^{9}-19a^{3}z^{7}+4a^{3}z^{5}+z^{5}a^{-3}+9a^{3}z^{3}-2z^{3}a^{-3}-4a^{3}z+za^{-3}+2a^{2}z^{10}+9a^{2}z^{8}-25a^{2}z^{6}+3z^{6}a^{-2}+21a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+2z^{2}a^{-2}+5az^{9}-2az^{7}+5z^{7}a^{-1}-8az^{5}-7z^{5}a^{-1}+15az^{3}+4z^{3}a^{-1}-8az-3za^{-1}+az^{-1}+a^{-1}z^{-1}+6z^{8}-7z^{6}+3z^{4}-1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

5

1

4

2

7

2

-5

0

11

5

6

-2

11

9

-2

-4

10

9

1

-6

8

11

3

-8

6

10

-4

-10

3

8

5

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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