L11a65

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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_20,12,21,11 X_22,16,5,15 X_16,22,17,21 X_8,20,9,19 X_18,10,19,9 X_10,18,11,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 8, -9, 4, -2, 11, -3, 5, -6, 9, -8, 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	$16q^{9/2}-15q^{7/2}+10q^{5/2}-7q^{3/2}+q^{21/2}-3q^{19/2}+7q^{17/2}-10q^{15/2}+14q^{13/2}-17q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-z^{3}a^{-7}-za^{-7}-a^{-7}z^{-1}-2z^{5}a^{-5}-4z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-3z^{9}a^{-5}-7z^{9}a^{-7}-4z^{9}a^{-9}-5z^{8}a^{-4}-9z^{8}a^{-6}-9z^{8}a^{-8}-5z^{8}a^{-10}-5z^{7}a^{-3}-4z^{7}a^{-5}+12z^{7}a^{-7}+8z^{7}a^{-9}-3z^{7}a^{-11}-3z^{6}a^{-2}+6z^{6}a^{-4}+27z^{6}a^{-6}+34z^{6}a^{-8}+15z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+22z^{5}a^{-5}-4z^{5}a^{-9}+8z^{5}a^{-11}+5z^{4}a^{-2}-32z^{4}a^{-6}-46z^{4}a^{-8}-16z^{4}a^{-10}+3z^{4}a^{-12}+2z^{3}a^{-1}-7z^{3}a^{-3}-29z^{3}a^{-5}-14z^{3}a^{-7}+2z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}-3z^{2}a^{-4}+15z^{2}a^{-6}+31z^{2}a^{-8}+12z^{2}a^{-10}-2z^{2}a^{-12}-za^{-1}+7za^{-3}+13za^{-5}+5za^{-7}+2a^{-4}-4a^{-6}-9a^{-8}-4a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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

2

18

5

1

-4

16

5

2

3

14

9

5

-4

12

8

5

3

10

8

9

1

8

7

8

-1

6

3

8

5

4

7

-3

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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