L11a39

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(3v^{2}-8v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-13q^{9/2}+16q^{7/2}-{\frac {1}{q^{7/2}}}-18q^{5/2}+{\frac {2}{q^{5/2}}}+18q^{3/2}-{\frac {6}{q^{3/2}}}+q^{15/2}-4q^{13/2}+8q^{11/2}-15{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+4z^{3}a^{-3}+a^{3}z+5za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-z^{3}a^{-1}-2az-2za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+6z^{8}a^{-6}-13z^{6}a^{-6}+5z^{4}a^{-6}-z^{2}a^{-6}+4z^{9}a^{-5}+z^{7}a^{-5}-24z^{5}a^{-5}+24z^{3}a^{-5}-9za^{-5}+a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-34z^{6}a^{-4}+26z^{4}a^{-4}-7z^{2}a^{-4}+a^{-4}+7z^{9}a^{-3}-4z^{7}a^{-3}+a^{3}z^{5}-23z^{5}a^{-3}-3a^{3}z^{3}+36z^{3}a^{-3}+3a^{3}z-17za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+z^{10}a^{-2}+10z^{8}a^{-2}+2a^{2}z^{6}-27z^{6}a^{-2}-3a^{2}z^{4}+29z^{4}a^{-2}-12z^{2}a^{-2}+a^{2}+3a^{-2}+3z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-2az^{5}-12z^{5}a^{-1}-3az^{3}+19z^{3}a^{-1}+3az-10za^{-1}-az^{-1}+a^{-1}z^{-1}+4z^{8}-5z^{6}+7z^{4}-7z^{2}+2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

3

12

5

1

-4

10

8

3

5

8

5

-3

6

10

8

2

4

8

0

2

7

10

-3

0

5

10

5

-2

1

5

-4

1

5

4

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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