L11a1

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,6,11,5 X₈₄₉₃ X_20,12,21,11 X_22,17,5,18 X_18,21,19,22 X_12,20,13,19 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, 9, -6, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-5v^{3}+7v^{2}-5v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+15q^{7/2}-21q^{5/2}+24q^{3/2}-25{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {16}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3az^{3}-4z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-az+2za^{-3}-za^{-5}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-7az^{9}-13z^{9}a^{-1}-6z^{9}a^{-3}-9a^{2}z^{8}-17z^{8}a^{-2}-9z^{8}a^{-4}-17z^{8}-5a^{3}z^{7}+5az^{7}+12z^{7}a^{-1}-6z^{7}a^{-3}-8z^{7}a^{-5}-a^{4}z^{6}+20a^{2}z^{6}+36z^{6}a^{-2}+9z^{6}a^{-4}-4z^{6}a^{-6}+44z^{6}+10a^{3}z^{5}+14az^{5}+16z^{5}a^{-1}+25z^{5}a^{-3}+12z^{5}a^{-5}-z^{5}a^{-7}+a^{4}z^{4}-11a^{2}z^{4}-19z^{4}a^{-2}-z^{4}a^{-4}+5z^{4}a^{-6}-25z^{4}-4a^{3}z^{3}-9az^{3}-14z^{3}a^{-1}-18z^{3}a^{-3}-8z^{3}a^{-5}+z^{3}a^{-7}+2z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}-a^{3}z-4az-2za^{-1}+3za^{-3}+2za^{-5}+1+a^{3}z^{-1}+2az^{-1}+2a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

9

3

-6

6

12

6

4

12

9

-3

2

13

12

1

0

10

14

4

-2

6

11

-5

-4

4

10

6

-6

1

6

-5

-8

4

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$