L11a74

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-8t(1)t(2)^{3}+12t(2)^{3}+12t(1)t(2)^{2}-8t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+8{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {16}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {15}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4za^{7}-3a^{7}z^{-1}+5z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-4z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}-z^{5}a+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+13a^{8}z^{4}-6a^{8}z^{2}+a^{8}+3a^{7}z^{9}-4a^{7}z^{7}-14a^{7}z^{5}+28a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-42a^{6}z^{6}+46a^{6}z^{4}-18a^{6}z^{2}+3a^{6}+8a^{5}z^{9}-9a^{5}z^{7}-30a^{5}z^{5}+50a^{5}z^{3}-25a^{5}z+4a^{5}z^{-1}+a^{4}z^{10}+17a^{4}z^{8}-52a^{4}z^{6}+45a^{4}z^{4}-15a^{4}z^{2}+2a^{4}+5a^{3}z^{9}+7a^{3}z^{7}-39a^{3}z^{5}+38a^{3}z^{3}-15a^{3}z+2a^{3}z^{-1}+10a^{2}z^{8}-13a^{2}z^{6}+4a^{2}z^{4}+z^{4}a^{-2}-2a^{2}z^{2}+a^{2}+11az^{7}-15az^{5}+4z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}-2az+8z^{6}-7z^{4}+z^{2}$ (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

6

1

-1

4

3

2

5

1

-4

0

8

3

5

-2

9

6

-3

-4

10

7

3

-6

8

9

1

-8

7

10

-3

-10

5

9

4

-12

2

6

-4

-14

1

5

4

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$