L11a411

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{2}w^{2}-2uv^{2}w-4uvw^{2}+5uvw-2uv+2uw^{2}-4uw+2u-2v^{2}w^{2}+4v^{2}w-2v^{2}+2vw^{2}-5vw+4v+2w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+10q^{5}-13q^{4}+15q^{3}-14q^{2}+13q-8+6q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}-5z^{2}+2a^{2}+a^{-2}+2a^{-4}-a^{-6}-4+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+6z^{8}a^{-6}+2z^{8}+2az^{7}-z^{7}a^{-1}-15z^{7}a^{-3}-7z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-27z^{6}a^{-4}-16z^{6}a^{-6}+3z^{6}a^{-8}-5az^{5}-2z^{5}a^{-1}+18z^{5}a^{-3}+3z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+7z^{4}a^{-2}+41z^{4}a^{-4}+19z^{4}a^{-6}-6z^{4}a^{-8}-13z^{4}+az^{3}-6z^{3}a^{-1}-8z^{3}a^{-3}+8z^{3}a^{-5}+7z^{3}a^{-7}-2z^{3}a^{-9}+6a^{2}z^{2}-3z^{2}a^{-2}-22z^{2}a^{-4}-9z^{2}a^{-6}+z^{2}a^{-8}+15z^{2}+4az+6za^{-1}-4za^{-5}-2za^{-7}-4a^{2}-2a^{-2}+4a^{-4}+2a^{-6}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

2

13

4

1

-3

11

6

2

4

9

7

4

-3

7

8

6

2

5

7

8

1

3

6

7

-1

1

4

9

5

-1

2

4

-2

-3

1

5

4

-5

1

-1

-7

1

Integral Khovanov Homology

(db, data source)

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