L11a501

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,5,15,6 X_10,3,11,4 X_4,13,5,14 X₂₇₃₈ X_6,9,1,10 X_18,12,19,11 X_12,18,7,17 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, 9, -11, 8, -7, 10, -9, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}vw^{3}-3u^{2}vw^{2}+3u^{2}vw-u^{2}v-u^{2}w^{3}+2u^{2}w^{2}-2u^{2}w+uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-uv^{2}-2uvw^{3}+6uvw^{2}-6uvw+2uv+uw^{3}-3uw^{2}+3uw-u+2v^{2}w^{2}-2v^{2}w+v^{2}+vw^{3}-3vw^{2}+3vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-16q+19-18q^{-1}+17q^{-2}-11q^{-3}+8q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}-z^{2}a^{-4}+2a^{4}z^{-2}+3a^{4}-a^{-4}-a^{2}z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-7a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-9a^{2}+a^{-2}-z^{6}-z^{4}+2z^{2}+4z^{-2}+6$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+12a^{2}z^{8}+5z^{8}a^{-2}+11z^{8}+3a^{5}z^{7}-2a^{3}z^{7}-2az^{7}+8z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-19a^{4}z^{6}-39a^{2}z^{6}-2z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}-7a^{5}z^{5}-17a^{3}z^{5}-29az^{5}-27z^{5}a^{-1}-7z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+26a^{4}z^{4}+51a^{2}z^{4}-6z^{4}a^{-2}-5z^{4}a^{-4}+21z^{4}+3a^{5}z^{3}+28a^{3}z^{3}+50az^{3}+31z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-23a^{4}z^{2}-43a^{2}z^{2}+5z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}-19a^{3}z-35az-19za^{-1}-2za^{-3}+za^{-5}+11a^{4}+22a^{2}-a^{-4}+13+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

5

1

-4

5

7

2

5

3

9

5

-4

1

10

7

3

-1

10

11

1

-3

7

8

-1

-5

5

11

6

-7

3

6

-3

-9

5

-11

1

3

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$