L11n26

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₃₈₄₉ X_13,22,14,5 X_21,14,22,15 X_11,18,12,19 X_9,20,10,21 X_19,10,20,11 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4t(1)t(2)^{3}-4t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-4t(2)+4}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+z^{3}a^{11}+4za^{11}+a^{11}z^{-1}-z^{5}a^{9}-3z^{3}a^{9}+2a^{9}z^{-1}-2z^{5}a^{7}-7z^{3}a^{7}-5za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{14}z^{4}-4a^{14}z^{2}+4a^{14}+a^{13}z^{5}-a^{13}z^{3}-a^{13}z-a^{13}z^{-1}+a^{12}z^{8}-6a^{12}z^{6}+18a^{12}z^{4}-21a^{12}z^{2}+9a^{12}+a^{11}z^{9}-5a^{11}z^{7}+11a^{11}z^{5}-4a^{11}z^{3}-3a^{11}z-a^{11}z^{-1}+3a^{10}z^{8}-13a^{10}z^{6}+24a^{10}z^{4}-14a^{10}z^{2}+4a^{10}+a^{9}z^{9}-2a^{9}z^{7}-a^{9}z^{5}+11a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}+2a^{8}z^{8}-5a^{8}z^{6}+2a^{8}z^{4}+4a^{8}z^{2}-2a^{8}+3a^{7}z^{7}-10a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-3a^{5}z^{3}+a^{5}z$ (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

4

2

-14

2

4

2

-16

4

3

1

-18

2

-20

1

4

-3

-22

0

-24

1

-1

Integral Khovanov Homology

(db, data source)

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