L11a78

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+15q^{7/2}-20q^{5/2}+22q^{3/2}-22{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-9z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+8az-9za^{-1}+3za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-5z^{4}a^{-6}+10z^{7}a^{-5}-15z^{5}a^{-5}+5z^{3}a^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-10z^{6}a^{-4}-3a^{4}z^{4}-4z^{4}a^{-4}+3a^{4}z^{2}+5z^{2}a^{-4}-a^{4}-a^{-4}+5z^{9}a^{-3}+3a^{3}z^{7}+11z^{7}a^{-3}-8a^{3}z^{5}-35z^{5}a^{-3}+8a^{3}z^{3}+23z^{3}a^{-3}-4a^{3}z-6za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+4a^{2}z^{8}+18z^{8}a^{-2}-5a^{2}z^{6}-32z^{6}a^{-2}-5a^{2}z^{4}+6z^{4}a^{-2}+10a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+3az^{9}+8z^{9}a^{-1}+5az^{7}+3z^{7}a^{-1}-25az^{5}-36z^{5}a^{-1}+28az^{3}+38z^{3}a^{-1}-15az-17za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{10}+12z^{8}-23z^{6}+3z^{4}+14z^{2}-7$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

4

-4

10

6

1

5

8

9

4

-5

6

11

6

5

4

11

9

-2

2

11

0

8

13

5

-2

5

9

-4

2

8

6

-6

1

5

-4

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$