L11a83

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_20,10,21,9 X_22,18,5,17 X_18,22,19,21 X_8,20,9,19 X_14,12,15,11 X_10,16,11,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 4, -9, 8, -2, 11, -8, 9, -3, 5, -6, 7, -4, 6, -5}

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

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

3

18

5

1

-4

16

6

3

14

9

5

-4

12

8

6

2

10

8

9

1

8

6

8

-2

6

3

8

5

4

3

6

-3

2

1

5

4

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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