L11a160

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_16,14,17,13 X_14,5,15,6 X_4,15,5,16 X_20,17,21,18 X_18,11,19,12 X_12,19,13,20 X_6,22,1,21
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8, 11, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-8q^{3/2}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-9za^{3}-2a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+4za-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-6a^{6}z^{6}-3a^{6}z^{4}+7a^{6}z^{2}-3a^{6}+3a^{5}z^{9}+3a^{5}z^{7}-21a^{5}z^{5}+23a^{5}z^{3}-12a^{5}z+3a^{5}z^{-1}+a^{4}z^{10}+10a^{4}z^{8}-22a^{4}z^{6}+9a^{4}z^{4}+4a^{4}z^{2}-3a^{4}+7a^{3}z^{9}-27a^{3}z^{5}+z^{5}a^{-3}+33a^{3}z^{3}-z^{3}a^{-3}-15a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}+13a^{2}z^{8}-25a^{2}z^{6}+4z^{6}a^{-2}+11a^{2}z^{4}-6z^{4}a^{-2}+2a^{2}z^{2}+2z^{2}a^{-2}+4az^{9}+7az^{7}+7z^{7}a^{-1}-26az^{5}-11z^{5}a^{-1}+24az^{3}+6z^{3}a^{-1}-8az-2za^{-1}+7z^{8}-6z^{6}-4z^{4}+4z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

5

1

4

2

8

3

-5

0

10

5

-2

10

9

-1

-4

10

9

1

-6

7

11

4

-8

5

9

-4

-10

2

7

5

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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