L11n123

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_9,16,10,17 X_20,11,21,12 X_8,21,9,22 X_18,7,19,8 X_12,19,13,20 X_15,10,16,11 X_22,17,5,18 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, -3, 8, 4, -7, 11, -2, -8, 3, 9, -6, 7, -4, 5, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+3t(1)t(2)^{4}-3t(2)^{4}-4t(1)t(2)^{3}+4t(2)^{3}+4t(1)t(2)^{2}-4t(2)^{2}-3t(1)t(2)+3t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {9}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{11}-2a^{11}z^{-1}+4z^{3}a^{9}+10za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-11z^{3}a^{7}-11za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-3z^{5}a^{13}+3z^{3}a^{13}-z^{8}a^{12}+z^{6}a^{12}-6z^{4}a^{12}+6z^{2}a^{12}-z^{9}a^{11}-z^{5}a^{11}-2z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-5z^{8}a^{10}+10z^{6}a^{10}-8z^{4}a^{10}-4z^{2}a^{10}+5a^{10}-z^{9}a^{9}-6z^{7}a^{9}+20z^{5}a^{9}-25z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-4z^{8}a^{8}+6z^{6}a^{8}+2z^{4}a^{8}-8z^{2}a^{8}+5a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-18z^{3}a^{7}+11za^{7}-3a^{7}z^{-1}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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

3

1

-2

-8

4

-10

4

3

-1

-12

7

4

3

-14

4

5

1

-16

5

6

-1

-18

2

4

2

-20

1

5

-4

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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