L11n312

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_9,20,10,21 X_7,16,8,17 X_13,18,14,19 X_19,14,20,15 X_15,22,16,11 X_17,10,18,5 X_21,8,22,9 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -3, 8}, {-11, 2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{3}-2uv^{2}w^{2}+2uv^{2}w-uv^{2}+uvw^{4}-2uvw^{3}+3uvw^{2}-3uvw+uv+uw-v^{2}w^{3}-vw^{4}+3vw^{3}-3vw^{2}+2vw-v+w^{4}-2w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-8q^{-3}+12q^{-4}-11q^{-5}+11q^{-6}-8q^{-7}+5q^{-8}-2q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+4a^{8}z^{-2}+7a^{8}-z^{6}a^{6}-4z^{4}a^{6}-9z^{2}a^{6}-5a^{6}z^{-2}-12a^{6}-z^{6}a^{4}-2z^{4}a^{4}+2z^{2}a^{4}+2a^{4}z^{-2}+5a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$3z^{3}a^{11}-4za^{11}+a^{11}z^{-1}+z^{6}a^{10}+4z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+5z^{7}a^{9}-15z^{5}a^{9}+28z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+6z^{8}a^{8}-21z^{6}a^{8}+39z^{4}a^{8}-32z^{2}a^{8}-4a^{8}z^{-2}+15a^{8}+2z^{9}a^{7}+4z^{7}a^{7}-27z^{5}a^{7}+46z^{3}a^{7}-33za^{7}+9a^{7}z^{-1}+10z^{8}a^{6}-33z^{6}a^{6}+44z^{4}a^{6}-37z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+2z^{9}a^{5}+2z^{7}a^{5}-20z^{5}a^{5}+25z^{3}a^{5}-18za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-10z^{6}a^{4}+6z^{4}a^{4}-8z^{2}a^{4}-2a^{4}z^{-2}+8a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+4z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

5

1

4

-5

5

4

-1

-7

7

3

4

-9

5

6

1

-11

6

0

-13

2

5

3

-15

3

6

-3

-17

3

-19

2

-2

Integral Khovanov Homology

(db, data source)

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