L11n306

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{4}-uv^{2}w^{3}+uv^{2}w^{2}-uv^{2}w+uvw-uv+u-v^{2}w^{4}+vw^{4}-vw^{3}+w^{3}-w^{2}+w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-3q^{-8}+4q^{-7}-4q^{-6}+5q^{-5}-3q^{-4}+4q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}z^{-2}-2a^{10}+a^{8}z^{6}+6a^{8}z^{4}+12a^{8}z^{2}+4a^{8}z^{-2}+11a^{8}-a^{6}z^{8}-7a^{6}z^{6}-18a^{6}z^{4}-24a^{6}z^{2}-5a^{6}z^{-2}-18a^{6}+a^{4}z^{6}+6a^{4}z^{4}+12a^{4}z^{2}+2a^{4}z^{-2}+9a^{4}$ (db)
Kauffman polynomial	$za^{13}+2z^{2}a^{12}-a^{12}+z^{5}a^{11}-z^{3}a^{11}-2za^{11}+a^{11}z^{-1}+3z^{6}a^{10}-10z^{4}a^{10}+6z^{2}a^{10}-a^{10}z^{-2}+5z^{7}a^{9}-23z^{5}a^{9}+31z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+4z^{8}a^{8}-19z^{6}a^{8}+27z^{4}a^{8}-21z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+z^{9}a^{7}+z^{7}a^{7}-25z^{5}a^{7}+50z^{3}a^{7}-35za^{7}+9a^{7}z^{-1}+5z^{8}a^{6}-29z^{6}a^{6}+55z^{4}a^{6}-46z^{2}a^{6}-5a^{6}z^{-2}+22a^{6}+z^{9}a^{5}-4z^{7}a^{5}-z^{5}a^{5}+18z^{3}a^{5}-19za^{5}+5a^{5}z^{-1}+z^{8}a^{4}-7z^{6}a^{4}+18z^{4}a^{4}-21z^{2}a^{4}-2a^{4}z^{-2}+11a^{4}$ (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

-3

0

-5

4

1

3

-7

1

2

1

-9

4

2

-11

1

2

1

-13

3

0

-15

1

2

1

-17

2

3

-1

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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