L11a70

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+7t(2)^{4}+10t(1)t(2)^{3}-10t(2)^{3}-10t(1)t(2)^{2}+10t(2)^{2}+7t(1)t(2)-5t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {22}{q^{9/2}}}-{\frac {23}{q^{7/2}}}+{\frac {20}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {19}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-6za^{7}-3a^{7}z^{-1}+3z^{5}a^{5}+8z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+3a^{10}z^{2}-a^{10}+5a^{9}z^{7}-6a^{9}z^{5}+3a^{9}z^{3}-2a^{9}z+a^{9}z^{-1}+6a^{8}z^{8}-4a^{8}z^{6}-4a^{8}z^{4}+8a^{8}z^{2}-3a^{8}+5a^{7}z^{9}+2a^{7}z^{7}-17a^{7}z^{5}+24a^{7}z^{3}-13a^{7}z+3a^{7}z^{-1}+2a^{6}z^{10}+12a^{6}z^{8}-28a^{6}z^{6}+16a^{6}z^{4}+4a^{6}z^{2}-3a^{6}+12a^{5}z^{9}-14a^{5}z^{7}-15a^{5}z^{5}+30a^{5}z^{3}-18a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-42a^{4}z^{6}+26a^{4}z^{4}-2a^{4}z^{2}-2a^{4}+7a^{3}z^{9}-6a^{3}z^{7}-15a^{3}z^{5}+15a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+9a^{2}z^{8}-20a^{2}z^{6}+10a^{2}z^{4}-a^{2}z^{2}+5az^{7}-10az^{5}+4az^{3}+az+z^{6}-z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

4

0

6

1

-5

-2

10

4

6

-4

11

7

-4

-6

12

9

3

-8

10

11

1

-10

9

12

-3

-12

5

11

6

-14

2

8

-6

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$