L11a82

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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_14,17,15,18 X_20,10,21,9 X_18,21,19,22 X_8,16,9,15 X_10,20,11,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 4, -5, 8, -3, 5, -7, 9, -6, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{4}-10uv^{3}+14uv^{2}-6uv+u+v^{5}-6v^{4}+14v^{3}-10v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+3q^{5/2}-{\frac {22}{q^{5/2}}}-8q^{3/2}+{\frac {21}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-2z^{5}a^{3}-4z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}-z^{5}a+3z^{3}a+6za+3az^{-1}+2z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}-za^{-3}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-3a^{7}z+6a^{6}z^{8}-12a^{6}z^{6}+5a^{6}z^{4}-a^{6}z^{2}+a^{6}+4a^{5}z^{9}+5a^{5}z^{7}-34a^{5}z^{5}+36a^{5}z^{3}-15a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+17a^{4}z^{8}-47a^{4}z^{6}+40a^{4}z^{4}-14a^{4}z^{2}+2a^{4}+9a^{3}z^{9}-a^{3}z^{7}-40a^{3}z^{5}+z^{5}a^{-3}+53a^{3}z^{3}-2z^{3}a^{-3}-24a^{3}z+za^{-3}+4a^{3}z^{-1}+a^{2}z^{10}+19a^{2}z^{8}-51a^{2}z^{6}+3z^{6}a^{-2}+53a^{2}z^{4}-4z^{4}a^{-2}-24a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+5az^{9}+4az^{7}+6z^{7}a^{-1}-26az^{5}-9z^{5}a^{-1}+34az^{3}+7z^{3}a^{-1}-17az-4za^{-1}+3az^{-1}+a^{-1}z^{-1}+8z^{8}-14z^{6}+16z^{4}-11z^{2}+1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

6

1

5

2

8

2

-6

0

10

6

4

-2

12

9

-3

-4

10

9

1

-6

8

12

4

-8

6

10

-4

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$