L11a3

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,12,21,11 X_22,20,5,19 X_18,13,19,14 X_12,22,13,21 X_2,16,3,15
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 $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+7t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+16q^{7/2}-22q^{5/2}+24q^{3/2}-25{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z-za^{-3}-3a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-4z^{3}a^{-1}+2az+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-13z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-19z^{8}a^{-2}-11z^{8}a^{-4}-15z^{8}-4a^{3}z^{7}+4az^{7}+13z^{7}a^{-1}-5z^{7}a^{-3}-10z^{7}a^{-5}-a^{4}z^{6}+15a^{2}z^{6}+43z^{6}a^{-2}+12z^{6}a^{-4}-5z^{6}a^{-6}+42z^{6}+9a^{3}z^{5}+14az^{5}+16z^{5}a^{-1}+26z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-9a^{2}z^{4}-27z^{4}a^{-2}-z^{4}a^{-4}+5z^{4}a^{-6}-32z^{4}-6a^{3}z^{3}-16az^{3}-20z^{3}a^{-1}-15z^{3}a^{-3}-5z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}+2a^{3}z+4az+za^{-1}-za^{-3}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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

4

-4

10

6

1

5

8

10

4

-6

6

12

6

4

12

10

-2

2

13

12

1

0

9

14

5

-2

6

11

-5

-4

3

9

6

-6

1

6

-5

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

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