L11n362

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,6,19,5 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_14,18,15,17 X_22,16,17,15 X_16,22,5,21 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, -7, 6, 10, -9}, {4, -1, 2, -5, -6, 7, 11, -2, 3, -8, 9, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)t(2)^{2}+t(3)^{2}t(2)-t(3)t(2)+t(2)+t(3)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$3q^{7}-5q^{6}+10q^{5}-12q^{4}+14q^{3}-13q^{2}-q^{-2}+11q+4q^{-1}-7$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+4z^{2}a^{-4}-3z^{2}a^{-6}-z^{2}-a^{-2}+3a^{-4}-4a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$6z^{4}a^{-8}-14z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+3z^{7}a^{-7}-6z^{5}a^{-7}+10z^{3}a^{-7}-10za^{-7}+2a^{-7}z^{-1}+5z^{8}a^{-6}-16z^{6}a^{-6}+33z^{4}a^{-6}-33z^{2}a^{-6}-2a^{-6}z^{-2}+15a^{-6}+2z^{9}a^{-5}+3z^{7}a^{-5}-13z^{5}a^{-5}+20z^{3}a^{-5}-12za^{-5}+2a^{-5}z^{-1}+10z^{8}a^{-4}-25z^{6}a^{-4}+33z^{4}a^{-4}-24z^{2}a^{-4}-a^{-4}z^{-2}+9a^{-4}+2z^{9}a^{-3}+6z^{7}a^{-3}-18z^{5}a^{-3}+14z^{3}a^{-3}-3za^{-3}+5z^{8}a^{-2}-5z^{6}a^{-2}-z^{4}a^{-2}-3z^{2}a^{-2}+2a^{-2}+6z^{7}a^{-1}+az^{5}-10z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}-za^{-1}+4z^{6}-7z^{4}+2z^{2}+1$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

15

3

13

4

2

-2

11

6

1

5

9

6

4

-2

7

8

6

2

5

6

1

3

6

8

-2

1

3

7

4

-1

1

4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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