L11a376

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)\left(t(2)t(1)^{2}+t(2)^{2}t(1)-t(2)t(1)+t(1)+t(2)\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	${\frac {14}{q^{9/2}}}-{\frac {15}{q^{7/2}}}+{\frac {12}{q^{5/2}}}+q^{3/2}-{\frac {10}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {12}{q^{11/2}}}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-8za^{5}-a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-5z^{3}a^{3}-2za^{3}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+5z^{5}a^{9}-2z^{3}a^{9}+za^{9}-4z^{8}a^{8}+10z^{6}a^{8}-15z^{4}a^{8}+9z^{2}a^{8}-3z^{9}a^{7}+5z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-z^{10}a^{6}-5z^{8}a^{6}+19z^{6}a^{6}-29z^{4}a^{6}+14z^{2}a^{6}-a^{6}-6z^{9}a^{5}+15z^{7}a^{5}-18z^{5}a^{5}+13z^{3}a^{5}-8za^{5}+a^{5}z^{-1}-z^{10}a^{4}-5z^{8}a^{4}+18z^{6}a^{4}-18z^{4}a^{4}+7z^{2}a^{4}-3z^{9}a^{3}+4z^{7}a^{3}+3z^{5}a^{3}-2z^{3}a^{3}+za^{3}-4z^{8}a^{2}+10z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}-3z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+3z^{4}-2z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

4

1

-3

-2

6

2

4

-4

7

5

-2

-6

8

5

3

-8

6

7

1

-10

6

8

-2

-12

3

6

3

-14

2

6

-4

-16

1

4

3

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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