L11a96

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Link Presentations

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

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

3

1

-2

10

6

2

4

8

6

3

-3

6

8

6

2

4

6

0

2

6

8

-2

0

5

8

3

-2

1

4

-3

-4

1

5

4

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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