L11n303

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_19,11,20,22 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{2}t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-t(1)t(2)t(3)^{3}+t(2)t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+2t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}+t(3)^{2}-t(1)t(2)t(3)+t(2)t(3)-t(3)+t(1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+2q^{7}-3q^{6}+4q^{5}-4q^{4}+4q^{3}-2q^{2}+2q+1+q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}+z^{6}a^{-4}-8z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-19z^{2}a^{-2}+14z^{2}a^{-4}-3z^{2}a^{-6}+5z^{2}-17a^{-2}+13a^{-4}-3a^{-6}+7-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}a^{-4}+z^{8}a^{-6}+z^{8}+z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+2z^{7}a^{-7}-10z^{6}a^{-2}-6z^{6}a^{-4}-2z^{6}a^{-6}+2z^{6}a^{-8}-8z^{6}-9z^{5}a^{-1}-16z^{5}a^{-3}-14z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+31z^{4}a^{-2}+15z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}a^{-8}+21z^{4}+22z^{3}a^{-1}+41z^{3}a^{-3}+26z^{3}a^{-5}+4z^{3}a^{-7}-3z^{3}a^{-9}-39z^{2}a^{-2}-16z^{2}a^{-4}+3z^{2}a^{-6}+3z^{2}a^{-8}-23z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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

1

13

2

1

-1

11

2

1

9

2

0

7

1

3

2

0

5

2

4

2

3

1

2

1

0

1

3

-1

1

-3

1

-5

1

Integral Khovanov Homology

(db, data source)

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