L11a261

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_18,8,19,7 X_14,6,15,5 X_22,17,9,18 X_16,21,17,22 X_20,13,21,14 X_6,16,7,15 X_4,20,5,19 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, 4, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, -6, 5, -3, 9, -7, 6, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+t(1)t(2)-t(2)+1\right)\left(t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+4q^{5/2}-{\frac {21}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+7a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-3a^{5}z+6a^{3}z-2az-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-5a^{5}z^{9}-11a^{3}z^{9}-6az^{9}-5a^{6}z^{8}-6a^{4}z^{8}-10a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+11a^{5}z^{7}+24a^{3}z^{7}+2az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+13a^{6}z^{6}+22a^{4}z^{6}+25a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}-13a^{5}z^{5}-27a^{3}z^{5}+8az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-12a^{6}z^{4}-25a^{4}z^{4}-20a^{2}z^{4}+5z^{4}a^{-2}-5z^{4}-5a^{7}z^{3}+13a^{5}z^{3}+22a^{3}z^{3}-4az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}+6a^{6}z^{2}+14a^{4}z^{2}+8a^{2}z^{2}-z^{2}a^{-2}+z^{2}-6a^{5}z-11a^{3}z-3az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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

3

-3

4

6

1

5

2

8

3

-5

0

11

6

5

-2

11

10

-1

-4

10

9

1

-6

7

11

4

-8

5

10

-5

-10

2

7

5

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$