L11n308

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)t(3)^{4}-2t(2)^{2}t(3)^{3}+2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(1)t(3)-2t(1)t(2)t(3)+t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{7}+q^{6}-q^{4}+3q^{3}+q^{-3}-3q^{2}-2q^{-2}+5q+4q^{-1}-3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+9a^{-4}-2z^{4}a^{-2}+a^{2}z^{2}-8z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-11a^{-2}-z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{7}a^{-7}-6z^{5}a^{-7}+9z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+z^{8}a^{-6}-7z^{6}a^{-6}+13z^{4}a^{-6}-9z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+z^{7}a^{-5}-10z^{5}a^{-5}+23z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+2z^{8}a^{-4}-15z^{6}a^{-4}+33z^{4}a^{-4}-29z^{2}a^{-4}-4a^{-4}z^{-2}+15a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}-10z^{5}a^{-3}+37z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+4z^{8}a^{-2}+a^{2}z^{6}-23z^{6}a^{-2}-4a^{2}z^{4}+43z^{4}a^{-2}+3a^{2}z^{2}-38z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+20a^{-2}+z^{9}a^{-1}+2az^{7}-z^{7}a^{-1}-7az^{5}-13z^{5}a^{-1}+3az^{3}+26z^{3}a^{-1}-18za^{-1}+5a^{-1}z^{-1}+3z^{8}-14z^{6}+19z^{4}-15z^{2}-2z^{-2}+8$ (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

15

1

-1

13

0

11

1

9

2

1

-1

7

2

1

2

5

4

5

1

0

3

2

1

2

1

2

5

1

2

-1

2

1

-3

2

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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