L11n109

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,18,8,19 X_19,22,20,5 X_13,20,14,21 X_21,14,22,15 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, -4, 5, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{3}-uv^{2}-uv+u+v^{3}-v^{2}-v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2za^{11}+2a^{11}z^{-1}+za^{9}-a^{9}z^{-1}-z^{5}a^{7}-4z^{3}a^{7}-2za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+7z^{6}a^{14}-15z^{4}a^{14}+11z^{2}a^{14}-3a^{14}-z^{9}a^{13}+7z^{7}a^{13}-15z^{5}a^{13}+12z^{3}a^{13}-4za^{13}+a^{13}z^{-1}-2z^{8}a^{12}+15z^{6}a^{12}-34z^{4}a^{12}+28z^{2}a^{12}-7a^{12}-z^{9}a^{11}+8z^{7}a^{11}-20z^{5}a^{11}+21z^{3}a^{11}-10za^{11}+2a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-18z^{4}a^{10}+14z^{2}a^{10}-4a^{10}+2z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{6}a^{8}+4z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}+4z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-3za^{5}+a^{5}z^{-1}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

1

-10

1

-12

1

3

1

-14

1

-16

1

2

-1

-18

1

-20

1

0

-22

1

0

-24

0

-26

1

-1

Integral Khovanov Homology

(db, data source)

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