L11n163

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_17,13,18,12 X_14,5,15,6 X_4,13,5,14 X_11,19,12,18 X_22,19,7,20 X_20,15,21,16 X_16,21,17,22 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, 8, -9, -3, 6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}-3t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-3t(2)^{2}+2t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-za^{9}+z^{5}a^{7}+2z^{3}a^{7}+2za^{7}+a^{7}z^{-1}+z^{5}a^{5}+z^{3}a^{5}-za^{5}-a^{5}z^{-1}-2z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-2z^{2}a^{12}-3z^{7}a^{11}+10z^{5}a^{11}-9z^{3}a^{11}+2za^{11}-3z^{8}a^{10}+7z^{6}a^{10}-z^{4}a^{10}-2z^{2}a^{10}-z^{9}a^{9}-4z^{7}a^{9}+17z^{5}a^{9}-14z^{3}a^{9}+5za^{9}-5z^{8}a^{8}+9z^{6}a^{8}-z^{4}a^{8}-z^{2}a^{8}-z^{9}a^{7}-3z^{7}a^{7}+7z^{5}a^{7}-2z^{3}a^{7}-2za^{7}+a^{7}z^{-1}-2z^{8}a^{6}+z^{4}a^{6}-a^{6}-2z^{7}a^{5}-2za^{5}+a^{5}z^{-1}-z^{6}a^{4}-2z^{4}a^{4}+z^{2}a^{4}-3z^{3}a^{3}+3za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

3

1

-2

-6

4

1

3

-8

5

3

-2

-10

5

4

1

-12

4

5

1

-14

4

5

-1

-16

2

5

3

-18

1

3

-2

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$