L11a457

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(3)^{2}t(2)^{3}+t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(3)^{3}t(2)^{2}-4t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-3t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+3t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+2t(1)t(2)-5t(1)t(3)t(2)+4t(3)t(2)-t(2)+t(1)t(3)^{3}-t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+10q-15+21q^{-1}-22q^{-2}+23q^{-3}-19q^{-4}+14q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3a^{6}z^{2}+2a^{6}-3a^{4}z^{4}-2a^{4}z^{2}+a^{4}z^{-2}+a^{4}+a^{2}z^{6}-3a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-5a^{2}-2z^{4}+z^{-2}+3$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-5a^{8}z^{4}+3a^{8}z^{2}-a^{8}+5a^{7}z^{7}-5a^{7}z^{5}+a^{7}z+7a^{6}z^{8}-9a^{6}z^{6}+6a^{6}z^{4}+6a^{5}z^{9}-3a^{5}z^{7}-6a^{5}z^{5}+10a^{5}z^{3}-3a^{5}z+2a^{4}z^{10}+15a^{4}z^{8}-45a^{4}z^{6}+52a^{4}z^{4}-30a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+13a^{3}z^{9}-21a^{3}z^{7}+3a^{3}z^{5}+10a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+16a^{2}z^{8}-53a^{2}z^{6}+z^{6}a^{-2}+59a^{2}z^{4}-2z^{4}a^{-2}-39a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+7az^{9}-9az^{7}+4z^{7}a^{-1}-5az^{5}-8z^{5}a^{-1}+6az^{3}+4z^{3}a^{-1}-5az+2az^{-1}+8z^{8}-19z^{6}+16z^{4}-11z^{2}-z^{-2}+6$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

7

1

6

1

8

3

-5

-1

13

7

6

-3

12

11

-1

-5

11

10

1

-7

8

12

4

-9

6

11

-5

-11

2

9

7

-13

1

5

-4

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$