L11n42

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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 X₃₈₄₉ X_5,14,6,15 X_15,22,16,5 X_17,20,18,21 X_21,16,22,17 X_19,10,20,11 X_11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3t(1)t(2)^{3}-4t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-4t(2)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {2}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+z^{3}a^{11}+3za^{11}+2a^{11}z^{-1}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$3a^{14}z^{4}-8a^{14}z^{2}+3a^{14}+a^{13}z^{7}-a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+2a^{12}z^{8}-10a^{12}z^{6}+24a^{12}z^{4}-24a^{12}z^{2}+7a^{12}+a^{11}z^{9}-4a^{11}z^{7}+10a^{11}z^{5}-11a^{11}z^{3}+8a^{11}z-2a^{11}z^{-1}+3a^{10}z^{8}-13a^{10}z^{6}+25a^{10}z^{4}-16a^{10}z^{2}+4a^{10}+a^{9}z^{9}-4a^{9}z^{7}+9a^{9}z^{5}-7a^{9}z^{3}+4a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-2a^{8}z^{6}+2a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

2

-10

2

1

-1

-12

4

2

-14

3

0

-16

3

0

-18

1

3

2

-20

2

3

-1

-22

1

-24

2

-2

Integral Khovanov Homology

(db, data source)

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