L11a534

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(4)^{2}t(3)^{2}+t(2)t(4)^{2}t(3)^{2}-t(4)^{2}t(3)^{2}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-2t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}-3t(2)t(4)t(3)^{2}+2t(4)t(3)^{2}-t(3)^{2}-3t(1)t(4)^{2}t(3)+t(1)t(2)t(4)^{2}t(3)-2t(2)t(4)^{2}t(3)+2t(4)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-3t(2)t(3)+6t(1)t(4)t(3)-4t(1)t(2)t(4)t(3)+6t(2)t(4)t(3)-4t(4)t(3)+t(3)+2t(1)t(4)^{2}-t(1)t(2)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}+t(1)-t(1)t(2)+t(2)-3t(1)t(4)+2t(1)t(2)t(4)-2t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {21}{q^{7/2}}}+{\frac {14}{q^{5/2}}}-{\frac {10}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {14}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {23}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}+a^{9}z^{-3}-6z^{3}a^{7}-11za^{7}-9a^{7}z^{-1}-3a^{7}z^{-3}+3z^{5}a^{5}+8z^{3}a^{5}+13za^{5}+10a^{5}z^{-1}+3a^{5}z^{-3}+z^{5}a^{3}-2z^{3}a^{3}-6za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}-z^{3}a$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+9a^{11}z^{3}-6a^{11}z+2a^{11}z^{-1}+4a^{10}z^{8}-4a^{10}z^{6}-8a^{10}z^{4}+14a^{10}z^{2}-6a^{10}+3a^{9}z^{9}+7a^{9}z^{7}-33a^{9}z^{5}+42a^{9}z^{3}-a^{9}z^{-3}-30a^{9}z+11a^{9}z^{-1}+a^{8}z^{10}+13a^{8}z^{8}-24a^{8}z^{6}-4a^{8}z^{4}+28a^{8}z^{2}+3a^{8}z^{-2}-18a^{8}+8a^{7}z^{9}+7a^{7}z^{7}-56a^{7}z^{5}+74a^{7}z^{3}-3a^{7}z^{-3}-49a^{7}z+18a^{7}z^{-1}+a^{6}z^{10}+19a^{6}z^{8}-37a^{6}z^{6}+9a^{6}z^{4}+24a^{6}z^{2}+6a^{6}z^{-2}-21a^{6}+5a^{5}z^{9}+12a^{5}z^{7}-48a^{5}z^{5}+56a^{5}z^{3}-3a^{5}z^{-3}-35a^{5}z+14a^{5}z^{-1}+10a^{4}z^{8}-14a^{4}z^{6}+4a^{4}z^{4}+7a^{4}z^{2}+3a^{4}z^{-2}-9a^{4}+9a^{3}z^{7}-16a^{3}z^{5}+14a^{3}z^{3}-a^{3}z^{-3}-10a^{3}z+5a^{3}z^{-1}+4a^{2}z^{6}-4a^{2}z^{4}+az^{5}-az^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

7

1

6

-4

8

4

-4

-6

13

6

7

-8

11

0

-10

12

10

2

-12

8

14

6

-14

6

9

-3

-16

2

9

7

-18

1

5

-4

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$