L11n41

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,19 X₈₄₉₃ X_5,14,6,15 X_15,20,16,21 X_17,22,18,5 X_21,16,22,17 X_19,10,20,11 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{12}+a^{12}-2z^{3}a^{11}+za^{11}-z^{6}a^{10}+z^{4}a^{10}-2z^{2}a^{10}-3z^{7}a^{9}+10z^{5}a^{9}-12z^{3}a^{9}+4za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+9z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-3a^{8}-z^{9}a^{7}-3z^{7}a^{7}+23z^{5}a^{7}-27z^{3}a^{7}+8za^{7}+a^{7}z^{-1}-5z^{8}a^{6}+18z^{6}a^{6}-14z^{4}a^{6}+2z^{2}a^{6}-z^{9}a^{5}-z^{7}a^{5}+18z^{5}a^{5}-26z^{3}a^{5}+12za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-2z^{2}a^{4}+3a^{4}-z^{7}a^{3}+5z^{5}a^{3}-9z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

3

0

-8

4

2

-10

1

3

1

-12

4

0

-14

2

3

1

-16

1

4

-3

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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