L11n355

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_7,16,8,17 X_15,8,16,9 X_20,13,21,14 X_22,20,15,19 X_12,21,13,22 X_14,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(uv^{2}w-uv^{2}-2uvw+3uv+uw-u-v^{2}w+v^{2}+3vw-2v-w+1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{-7}-4q^{-6}+9q^{-5}-10q^{-4}+13q^{-3}-q^{2}-12q^{-2}+4q+10q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+z^{2}a^{6}-2a^{6}z^{-2}-2a^{6}-2z^{4}a^{4}-2z^{2}a^{4}+a^{4}z^{-2}+a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+a^{7}z^{7}+a^{7}z^{5}+a^{7}z^{3}-5a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-2a^{6}z^{6}+7a^{6}z^{4}-10a^{6}z^{2}-2a^{6}z^{-2}+6a^{6}+a^{5}z^{9}+4a^{5}z^{7}-7a^{5}z^{5}+7a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+6a^{4}z^{8}-7a^{4}z^{6}+2a^{4}z^{4}-a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+a^{3}z^{9}+9a^{3}z^{7}-20a^{3}z^{5}+11a^{3}z^{3}+4a^{2}z^{8}-a^{2}z^{6}-9a^{2}z^{4}+5a^{2}z^{2}-a^{2}+6az^{7}-11az^{5}+z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

1

4

1

-3

-1

6

3

-3

7

5

-2

-5

6

5

1

-7

5

8

3

-9

4

5

-1

-11

1

6

5

-13

1

3

-2

-15

2

Integral Khovanov Homology

(db, data source)

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