L11a456

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w^{2}-uv^{3}w+uv^{2}w^{3}-4uv^{2}w^{2}+5uv^{2}w-uv^{2}-2uvw^{3}+5uvw^{2}-4uvw+uv+uw^{3}-2uw^{2}+uw-v^{3}w^{2}+2v^{3}w-v^{3}-v^{2}w^{3}+4v^{2}w^{2}-5v^{2}w+2v^{2}+vw^{3}-5vw^{2}+4vw-v+w^{2}-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}-q^{-5}+7q^{4}+3q^{-4}-11q^{3}-7q^{-3}+17q^{2}+13q^{-2}-18q-16q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+2z^{2}a^{-4}+a^{-4}z^{-2}-a^{4}+2a^{-4}-z^{6}a^{-2}+2a^{2}z^{4}-3z^{4}a^{-2}+3a^{2}z^{2}-6z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-5a^{-2}-z^{6}-z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{6}a^{-6}-3z^{4}a^{-6}+2z^{2}a^{-6}+3z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}-2a^{5}z^{3}+5z^{3}a^{-5}+a^{5}z+5z^{8}a^{-4}+3a^{4}z^{6}-14z^{6}a^{-4}-5a^{4}z^{4}+16z^{4}a^{-4}+3a^{4}z^{2}-13z^{2}a^{-4}-a^{-4}z^{-2}-a^{4}+6a^{-4}+4z^{9}a^{-3}+5a^{3}z^{7}-5z^{7}a^{-3}-6a^{3}z^{5}-3z^{5}a^{-3}+a^{3}z^{3}+6z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+6a^{2}z^{8}+12z^{8}a^{-2}-7a^{2}z^{6}-41z^{6}a^{-2}+3a^{2}z^{4}+55z^{4}a^{-2}-a^{2}z^{2}-39z^{2}a^{-2}-2a^{-2}z^{-2}+13a^{-2}+4az^{9}+8z^{9}a^{-1}+2az^{7}-11z^{7}a^{-1}-12az^{5}+10az^{3}+8z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+z^{10}+13z^{8}-36z^{6}+44z^{4}-28z^{2}-z^{-2}+9$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

5

1

4

7

3

-4

5

10

4

6

3

8

7

-1

1

11

10

1

-1

8

11

3

-3

5

8

-3

-5

2

8

6

-7

1

5

-4

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$