L11a113

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+8t(1)t(2)^{3}-10t(2)^{3}-10t(1)t(2)^{2}+8t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {13}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-8q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-10za^{3}-4a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+4za+az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-7a^{3}z^{9}-4az^{9}-4a^{6}z^{8}-10a^{4}z^{8}-13a^{2}z^{8}-7z^{8}-3a^{7}z^{7}-4a^{5}z^{7}-a^{3}z^{7}-7az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+5a^{6}z^{6}+18a^{4}z^{6}+22a^{2}z^{6}-4z^{6}a^{-2}+6z^{6}+8a^{7}z^{5}+23a^{5}z^{5}+26a^{3}z^{5}+23az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+6a^{6}z^{4}+3a^{4}z^{4}-3a^{2}z^{4}+6z^{4}a^{-2}+3z^{4}-8a^{7}z^{3}-25a^{5}z^{3}-27a^{3}z^{3}-17az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-11a^{6}z^{2}-16a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+4a^{7}z+15a^{5}z+14a^{3}z+4az+za^{-1}+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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

5

1

4

2

8

3

-5

0

10

5

-2

11

9

-2

-4

9

0

-6

7

11

4

-8

6

9

-3

-10

2

8

6

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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