L11a474

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(2)^{2}t(3)^{3}-t(2)t(3)^{3}-t(2)^{2}t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}+t(2)^{2}t(3)-t(2)t(3)+t(3)+t(2)-1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-11q^{5}+13q^{4}-11q^{3}+11q^{2}-7q+5-2q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-5z^{2}a^{-6}-3a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+14z^{4}a^{-4}+17z^{2}a^{-4}+a^{-4}z^{-2}+9a^{-4}-2z^{6}a^{-2}-10z^{4}a^{-2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-10a^{-2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+2z^{8}a^{-2}+10z^{8}a^{-4}+9z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-28z^{7}a^{-3}-8z^{7}a^{-5}+10z^{7}a^{-7}-29z^{6}a^{-2}-54z^{6}a^{-4}-22z^{6}a^{-6}+9z^{6}a^{-8}-6z^{6}+15z^{5}a^{-1}+27z^{5}a^{-3}-13z^{5}a^{-5}-19z^{5}a^{-7}+6z^{5}a^{-9}+61z^{4}a^{-2}+76z^{4}a^{-4}+12z^{4}a^{-6}-13z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-5z^{3}a^{-1}+3z^{3}a^{-3}+19z^{3}a^{-5}+6z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-47z^{2}a^{-2}-44z^{2}a^{-4}-5z^{2}a^{-6}+5z^{2}a^{-8}-13z^{2}-4za^{-1}-9za^{-3}-6za^{-5}+za^{-9}+15a^{-2}+13a^{-4}+2a^{-6}-a^{-8}+6+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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

19

1

-1

17

2

15

4

1

-3

13

5

2

3

11

6

4

-2

9

7

5

2

7

6

8

2

5

0

3

4

8

4

1

3

-2

-1

1

4

3

-3

1

-1

-5

1

Integral Khovanov Homology

(db, data source)

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