L11a281

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-5t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-3t(1)^{3}t(2)^{3}+10t(1)^{2}t(2)^{3}-11t(1)t(2)^{3}+3t(2)^{3}+3t(1)^{3}t(2)^{2}-11t(1)^{2}t(2)^{2}+10t(1)t(2)^{2}-3t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-5t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}-q^{7/2}+{\frac {23}{q^{7/2}}}+4q^{5/2}-{\frac {27}{q^{5/2}}}-9q^{3/2}+{\frac {26}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+16{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{5}z+a^{5}z^{-1}+a^{3}z^{7}+2a^{3}z^{5}-a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}+2az-za^{-1}$ (db)
Kauffman polynomial	$-4a^{4}z^{10}-4a^{2}z^{10}-10a^{5}z^{9}-20a^{3}z^{9}-10az^{9}-10a^{6}z^{8}-11a^{4}z^{8}-12a^{2}z^{8}-11z^{8}-5a^{7}z^{7}+17a^{5}z^{7}+43a^{3}z^{7}+13az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+21a^{6}z^{6}+40a^{4}z^{6}+39a^{2}z^{6}-4z^{6}a^{-2}+17z^{6}+9a^{7}z^{5}-4a^{5}z^{5}-30a^{3}z^{5}-5az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+a^{8}z^{4}-10a^{6}z^{4}-28a^{4}z^{4}-33a^{2}z^{4}+5z^{4}a^{-2}-11z^{4}-3a^{7}z^{3}+a^{5}z^{3}+7a^{3}z^{3}-3az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}+5a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}+2z^{2}-a^{7}z+a^{5}z+2a^{3}z+az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

10

3

-7

0

13

6

7

-2

14

11

-3

-4

13

12

1

-6

10

14

4

-8

8

13

-5

-10

4

11

7

-12

1

7

-6

-14

4

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$