L11n29

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_13,22,14,5 X_15,10,16,11 X_21,14,22,15 X_11,20,12,21 X_17,2,18,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, -11, -2, 3, 10, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4t(1)t(2)^{3}-5t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-5t(2)+4}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {2}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {7}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {5}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+a^{11}z^{3}+3a^{11}z+a^{11}z^{-1}-a^{9}z^{5}-2a^{9}z^{3}+2a^{9}z+2a^{9}z^{-1}-2a^{7}z^{5}-7a^{7}z^{3}-6a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-3z^{4}a^{14}+8z^{2}a^{14}-4a^{14}-z^{7}a^{13}+z^{5}a^{13}+z^{3}a^{13}+a^{13}z^{-1}-2z^{8}a^{12}+9z^{6}a^{12}-23z^{4}a^{12}+26z^{2}a^{12}-9a^{12}-z^{9}a^{11}+2z^{7}a^{11}-2z^{5}a^{11}-3z^{3}a^{11}+2za^{11}+a^{11}z^{-1}-4z^{8}a^{10}+15z^{6}a^{10}-25z^{4}a^{10}+14z^{2}a^{10}-4a^{10}-z^{9}a^{9}+8z^{5}a^{9}-19z^{3}a^{9}+11za^{9}-2a^{9}z^{-1}-2z^{8}a^{8}+4z^{6}a^{8}-5z^{2}a^{8}+2a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-12z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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

-4

1

-6

2

1

-1

-8

2

-10

3

2

-1

-12

5

2

3

-14

3

4

1

-16

4

0

-18

1

3

2

-20

2

4

-2

-22

1

-24

2

-2

Integral Khovanov Homology

(db, data source)

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