L11n71

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_17,22,18,5 X_11,18,12,19 X_21,12,22,13 X_13,20,14,21 X_19,14,20,15 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{3}-uv^{2}-2uv+2u+2v^{3}-2v^{2}-v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2za^{11}+a^{11}z^{-1}+3za^{9}+2a^{9}z^{-1}-z^{5}a^{7}-5z^{3}a^{7}-5za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-2za^{5}$ (db)
Kauffman polynomial	$a^{14}z^{8}-7a^{14}z^{6}+16a^{14}z^{4}-14a^{14}z^{2}+4a^{14}+a^{13}z^{9}-6a^{13}z^{7}+10a^{13}z^{5}-5a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+3a^{12}z^{8}-20a^{12}z^{6}+41a^{12}z^{4}-33a^{12}z^{2}+9a^{12}+a^{11}z^{9}-5a^{11}z^{7}+3a^{11}z^{5}+5a^{11}z^{3}-2a^{11}z-a^{11}z^{-1}+2a^{10}z^{8}-13a^{10}z^{6}+22a^{10}z^{4}-13a^{10}z^{2}+4a^{10}+2a^{9}z^{7}-14a^{9}z^{5}+24a^{9}z^{3}-13a^{9}z+2a^{9}z^{-1}+a^{8}z^{6}-7a^{8}z^{4}+8a^{8}z^{2}-2a^{8}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{6}-4a^{6}z^{4}+2a^{6}z^{2}+a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

1

0

-10

1

-12

2

4

1

-14

1

2

-16

1

3

2

0

-18

2

1

-20

1

0

-22

1

2

-1

-24

0

-26

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$	$i=-2$
$r=-11$		${\mathbb {Z} }$
$r=-10$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }^{2}$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{4}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$