L11n456

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_15,18,16,11 X_9,21,10,20 X_13,19,14,22 X_21,15,22,14 X_19,5,20,10 X_17,8,18,9 X_7,16,8,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {-7, 4, -6, 5}, {10, -1, -9, 8, -4, 7}, {11, -2, -5, 6, -3, 9, -8, 3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-uvwx+uvw+uvx-uv-uw^{2}x^{2}+2uwx-uw-ux+u+vw^{2}x^{2}-vw^{2}x-vwx^{2}+2vwx-v-w^{2}x^{2}+w^{2}x+wx^{2}-wx}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-q^{5/2}-{\frac {7}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}+a^{5}z^{-3}-z^{5}a^{3}-4z^{3}a^{3}-9za^{3}-7a^{3}z^{-1}-3a^{3}z^{-3}+z^{5}a+7z^{3}a+11za+8az^{-1}+3az^{-3}-z^{3}a^{-1}-4za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-11a^{7}z^{5}+9a^{7}z^{3}-3a^{7}z+3a^{6}z^{8}-10a^{6}z^{6}+5a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{9}+a^{5}z^{7}-16a^{5}z^{5}+22a^{5}z^{3}-a^{5}z^{-3}-12a^{5}z+5a^{5}z^{-1}+4a^{4}z^{8}-16a^{4}z^{6}+16a^{4}z^{4}+3a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{9}-3a^{3}z^{7}-2a^{3}z^{5}+17a^{3}z^{3}-3a^{3}z^{-3}-21a^{3}z+12a^{3}z^{-1}+a^{2}z^{8}-5a^{2}z^{6}+4a^{2}z^{4}+15a^{2}z^{2}+6a^{2}z^{-2}-19a^{2}+z^{7}a^{-1}-4az^{5}-7z^{5}a^{-1}+18az^{3}+14z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-23az-11za^{-1}+12az^{-1}+5a^{-1}z^{-1}-4z^{4}+12z^{2}+3z^{-2}-10$ (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

3

4

χ

6

1

4

1

2

1

0

4

-2

2

4

1

-4

6

2

4

-6

3

6

1

2

-8

3

1

-10

2

4

2

-12

1

2

-1

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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