L11a438

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_10,4,11,3 X_14,6,15,5 X_22,12,13,11 X_12,14,5,13 X_4,21,1,22 X_20,17,21,18 X_16,7,17,8 X_8,20,9,19 X_18,10,19,9
Gauss code	{1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, {6, -4, 2, -9, 8, -11, 10, -8, 7, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+14q^{5}-17q^{4}+21q^{3}-20q^{2}+18q-12+8q^{-1}-4q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-9z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-6z^{2}a^{-2}+6z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}-a^{-4}+1+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+7z^{7}a^{-7}-10z^{5}a^{-7}+2z^{3}a^{-7}+8z^{8}a^{-6}-11z^{6}a^{-6}+3z^{4}a^{-6}+z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+6z^{9}a^{-5}-5z^{7}a^{-5}-z^{5}a^{-5}-z^{3}a^{-5}+4za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-35z^{6}a^{-4}+37z^{4}a^{-4}-11z^{2}a^{-4}+2a^{-4}z^{-2}-4a^{-4}+12z^{9}a^{-3}-28z^{7}a^{-3}+24z^{5}a^{-3}-7z^{3}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+10z^{8}a^{-2}+a^{2}z^{6}-41z^{6}a^{-2}-2a^{2}z^{4}+47z^{4}a^{-2}-16z^{2}a^{-2}+a^{-2}z^{-2}-a^{-2}+6z^{9}a^{-1}+4az^{7}-12z^{7}a^{-1}-10az^{5}+4z^{5}a^{-1}+4az^{3}+z^{3}a^{-1}+7z^{8}-20z^{6}+17z^{4}-6z^{2}+1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

5

1

-4

11

9

3

6

9

10

7

-3

7

11

7

4

5

9

10

1

3

9

11

-2

1

5

11

6

-1

3

7

-4

-3

1

5

4

-5

3

-3

-7

1

Integral Khovanov Homology

(db, data source)

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