L11a410

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}+t(1)t(3)^{4}-2t(1)t(2)t(3)^{4}+t(2)t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)-t(2)^{2}+t(1)-t(1)t(2)+2t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-12q^{5}+13q^{4}-12q^{3}+12q^{2}-7q+6-2q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-17z^{2}a^{-2}+17z^{2}a^{-4}-5z^{2}a^{-6}+4z^{2}-13a^{-2}+11a^{-4}-3a^{-6}+5-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}+6z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+9z^{6}a^{-8}-12z^{4}a^{-8}+3z^{2}a^{-8}+11z^{7}a^{-7}-23z^{5}a^{-7}+12z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+10z^{8}a^{-6}-26z^{6}a^{-6}+19z^{4}a^{-6}-12z^{2}a^{-6}-a^{-6}z^{-2}+5a^{-6}+5z^{9}a^{-5}-4z^{7}a^{-5}-30z^{5}a^{-5}+43z^{3}a^{-5}-21za^{-5}+5a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-62z^{6}a^{-4}+87z^{4}a^{-4}-52z^{2}a^{-4}-4a^{-4}z^{-2}+18a^{-4}+7z^{9}a^{-3}-24z^{7}a^{-3}+9z^{5}a^{-3}+31z^{3}a^{-3}-29za^{-3}+9a^{-3}z^{-1}+z^{10}a^{-2}+3z^{8}a^{-2}-33z^{6}a^{-2}+67z^{4}a^{-2}-53z^{2}a^{-2}-5a^{-2}z^{-2}+21a^{-2}+2z^{9}a^{-1}-9z^{7}a^{-1}+10z^{5}a^{-1}+5z^{3}a^{-1}-13za^{-1}+5a^{-1}z^{-1}+z^{8}-6z^{6}+14z^{4}-16z^{2}-2z^{-2}+9$ (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

19

1

-1

17

2

15

4

1

-3

13

5

2

3

11

7

4

-3

9

6

5

1

7

8

1

5

0

3

4

9

5

1

2

3

-1

1

5

4

-3

1

-1

-5

1

Integral Khovanov Homology

(db, data source)

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