L11n7

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,20,12,21 X_19,22,20,5 X_13,19,14,18 X_21,12,22,13 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, -8, 7, -10, 8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+4t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+4t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {5}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+z^{3}a^{9}+4za^{9}+3a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{12}+a^{12}-z^{5}a^{11}-3z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-3z^{6}a^{10}+5z^{4}a^{10}-6z^{2}a^{10}+2a^{10}-4z^{7}a^{9}+11z^{5}a^{9}-15z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-3z^{8}a^{8}+7z^{6}a^{8}-4z^{4}a^{8}+3z^{2}a^{8}-z^{9}a^{7}-3z^{7}a^{7}+21z^{5}a^{7}-27z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-5z^{8}a^{6}+19z^{6}a^{6}-20z^{4}a^{6}+9z^{2}a^{6}-2a^{6}-z^{9}a^{5}+14z^{5}a^{5}-23z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+9z^{6}a^{4}-11z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

3

2

-1

-8

4

2

-10

3

0

-12

4

0

-14

2

4

2

-16

2

3

-1

-18

2

-20

2

-2

Integral Khovanov Homology

(db, data source)

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