L11n295

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_20,16,21,15 X_14,8,15,7 X_21,10,22,5 X_11,19,12,18 X_9,17,10,16 X_17,11,18,22 X_8,19,9,20 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {-6, -2, 11, -4, 3, 7, -8, 6, 9, -3, -5, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-uv^{2}w-3uvw^{2}+4uvw-2uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+3v+w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+8q^{5}-10q^{4}+11q^{3}-8q^{2}+8q-3+2q^{-1}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+4z^{4}a^{-4}+8z^{2}a^{-4}+4a^{-4}z^{-2}+8a^{-4}-3z^{4}a^{-2}-9z^{2}a^{-2}-5a^{-2}z^{-2}-10a^{-2}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+z^{2}a^{-8}+5z^{7}a^{-7}-13z^{5}a^{-7}+11z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+4z^{8}a^{-6}-9z^{6}a^{-6}+9z^{4}a^{-6}-6z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+z^{9}a^{-5}+6z^{7}a^{-5}-23z^{5}a^{-5}+32z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+6z^{8}a^{-4}-18z^{6}a^{-4}+30z^{4}a^{-4}-24z^{2}a^{-4}-4a^{-4}z^{-2}+12a^{-4}+z^{9}a^{-3}+2z^{7}a^{-3}-10z^{5}a^{-3}+24z^{3}a^{-3}-24za^{-3}+9a^{-3}z^{-1}+2z^{8}a^{-2}-6z^{6}a^{-2}+18z^{4}a^{-2}-25z^{2}a^{-2}-5a^{-2}z^{-2}+15a^{-2}+z^{7}a^{-1}-z^{5}a^{-1}+5z^{3}a^{-1}-11za^{-1}+5a^{-1}z^{-1}+3z^{4}-8z^{2}-2z^{-2}+7$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

4

1

-3

11

4

2

9

6

4

-2

7

5

4

1

5

4

7

3

4

0

1

6

5

-1

1

2

-1

-3

2

Integral Khovanov Homology

(db, data source)

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