L11a64

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-2t(2)^{3}+3t(2)^{2}-2t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+7q^{7/2}-5q^{5/2}+2q^{3/2}+q^{23/2}-3q^{21/2}+7q^{19/2}-10q^{17/2}+13q^{15/2}-14q^{13/2}+13q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-4z^{3}a^{-5}-6z^{3}a^{-7}+3z^{3}a^{-9}+4za^{-3}-7za^{-7}+3za^{-9}+a^{-3}z^{-1}+a^{-5}z^{-1}-4a^{-7}z^{-1}+2a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}-z^{2}a^{-14}+3z^{5}a^{-13}-2z^{3}a^{-13}+6z^{6}a^{-12}-8z^{4}a^{-12}+6z^{2}a^{-12}-2a^{-12}+7z^{7}a^{-11}-9z^{5}a^{-11}+4z^{3}a^{-11}+za^{-11}+6z^{8}a^{-10}-7z^{6}a^{-10}+4z^{2}a^{-10}-a^{-10}+4z^{9}a^{-9}-6z^{7}a^{-9}+9z^{5}a^{-9}-15z^{3}a^{-9}+8za^{-9}-2a^{-9}z^{-1}+z^{10}a^{-8}+6z^{8}a^{-8}-21z^{6}a^{-8}+24z^{4}a^{-8}-19z^{2}a^{-8}+6a^{-8}+6z^{9}a^{-7}-19z^{7}a^{-7}+25z^{5}a^{-7}-24z^{3}a^{-7}+13za^{-7}-4a^{-7}z^{-1}+z^{10}a^{-6}+2z^{8}a^{-6}-16z^{6}a^{-6}+23z^{4}a^{-6}-16z^{2}a^{-6}+5a^{-6}+2z^{9}a^{-5}-5z^{7}a^{-5}-z^{5}a^{-5}+5z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-8z^{6}a^{-4}+8z^{4}a^{-4}-a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+8z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

5

1

-4

18

5

2

3

16

8

5

-3

14

6

5

1

12

7

8

1

10

5

6

-1

8

2

7

5

6

3

5

-2

4

1

4

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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