L11n447

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_19,17,20,22 X_21,9,22,16 X_15,21,16,20 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {-9, 5, -6, 8, -7, 6}, {11, -2, -5, 9, 4, -3, -8, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)(x-1)\left(-uv+ux^{2}-ux+u+vx^{2}-vx+v-x^{2}\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+2{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+a^{9}z^{-3}-4za^{7}-7a^{7}z^{-1}-3a^{7}z^{-3}+5z^{3}a^{5}+13za^{5}+12a^{5}z^{-1}+3a^{5}z^{-3}-2z^{5}a^{3}-7z^{3}a^{3}-11za^{3}-7a^{3}z^{-1}-a^{3}z^{-3}+2z^{3}a+2za+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+10a^{9}z^{3}-a^{9}z^{-3}-10a^{9}z+5a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+5a^{8}z^{4}+6a^{8}z^{2}+3a^{8}z^{-2}-9a^{8}+a^{7}z^{9}+5a^{7}z^{7}-34a^{7}z^{5}+55a^{7}z^{3}-3a^{7}z^{-3}-38a^{7}z+14a^{7}z^{-1}+8a^{6}z^{8}-22a^{6}z^{6}+a^{6}z^{4}+27a^{6}z^{2}+6a^{6}z^{-2}-21a^{6}+a^{5}z^{9}+15a^{5}z^{7}-65a^{5}z^{5}+83a^{5}z^{3}-3a^{5}z^{-3}-54a^{5}z+18a^{5}z^{-1}+6a^{4}z^{8}-8a^{4}z^{6}-18a^{4}z^{4}+33a^{4}z^{2}+3a^{4}z^{-2}-18a^{4}+11a^{3}z^{7}-35a^{3}z^{5}+45a^{3}z^{3}-a^{3}z^{-3}-31a^{3}z+11a^{3}z^{-1}+7a^{2}z^{6}-14a^{2}z^{4}+15a^{2}z^{2}-6a^{2}+az^{5}+7az^{3}-5az+2az^{-1}+3z^{2}-1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

-2

0

4

-2

6

4

-2

-4

5

3

1

3

-6

4

6

2

-8

8

5

3

-10

5

10

5

-12

1

2

-1

-14

1

5

4

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-8$		${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$		${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{8}$
$r=-3$		${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$		${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$		${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$