L11n307

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 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_11,1,12,4
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 $u$ , $v$ , $w$ , ...)	${\frac {-3uv^{2}w^{2}+3uv^{2}w-uv^{2}-uvw^{3}+3uvw^{2}-2uvw+uw^{3}-uw^{2}+v^{2}w^{2}-v^{2}w+2vw^{3}-3vw^{2}+vw+w^{4}-3w^{3}+3w^{2}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{10}+3q^{9}-6q^{8}+8q^{7}-9q^{6}+11q^{5}-8q^{4}+8q^{3}-4q^{2}+2q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-3z^{4}a^{-6}+2z^{2}a^{-2}+2z^{2}a^{-4}-9z^{2}a^{-6}+4z^{2}a^{-8}+a^{-2}+5a^{-4}-12a^{-6}+7a^{-8}-a^{-10}+2a^{-4}z^{-2}-5a^{-6}z^{-2}+4a^{-8}z^{-2}-a^{-10}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-11}-4z^{5}a^{-11}+6z^{3}a^{-11}-4za^{-11}+a^{-11}z^{-1}+3z^{8}a^{-10}-12z^{6}a^{-10}+14z^{4}a^{-10}-6z^{2}a^{-10}-a^{-10}z^{-2}+3a^{-10}+2z^{9}a^{-9}-z^{7}a^{-9}-21z^{5}a^{-9}+36z^{3}a^{-9}-19za^{-9}+5a^{-9}z^{-1}+10z^{8}a^{-8}-39z^{6}a^{-8}+48z^{4}a^{-8}-32z^{2}a^{-8}-4a^{-8}z^{-2}+15a^{-8}+2z^{9}a^{-7}+5z^{7}a^{-7}-38z^{5}a^{-7}+52z^{3}a^{-7}-33za^{-7}+9a^{-7}z^{-1}+7z^{8}a^{-6}-24z^{6}a^{-6}+35z^{4}a^{-6}-37z^{2}a^{-6}-5a^{-6}z^{-2}+20a^{-6}+7z^{7}a^{-5}-20z^{5}a^{-5}+25z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+3z^{6}a^{-4}+z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+z^{5}a^{-3}+3z^{3}a^{-3}+3z^{2}a^{-2}-a^{-2}$ (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		\

0

1

2

3

4

5

6

7

8

9

χ

21

1

-1

19

2

17

4

1

-3

15

4

2

13

6

5

-1

11

5

3

2

9

3

6

3

7

5

0

5

4

3

2

4

-2

1

2

Integral Khovanov Homology

(db, data source)

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