L10a166

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_20,13,15,14 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_18,11,19,12 X_12,19,13,20 X_14,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 8, -9, 3, -10}, {7, -6, 10, -8, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-3t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-3t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {2}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {12}{q^{17/2}}}+{\frac {11}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {8}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-1}+a^{13}z^{-3}-4za^{11}-8a^{11}z^{-1}-3a^{11}z^{-3}+6z^{3}a^{9}+17za^{9}+13a^{9}z^{-1}+3a^{9}z^{-3}-3z^{5}a^{7}-11z^{3}a^{7}-13za^{7}-6a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+3z^{3}a^{15}-3za^{15}+a^{15}z^{-1}-2z^{6}a^{14}+3z^{4}a^{14}-a^{14}-3z^{7}a^{13}+4z^{5}a^{13}-2z^{3}a^{13}+3za^{13}-3a^{13}z^{-1}+a^{13}z^{-3}-2z^{8}a^{12}-4z^{6}a^{12}+15z^{4}a^{12}-16z^{2}a^{12}-3a^{12}z^{-2}+11a^{12}-z^{9}a^{11}-6z^{7}a^{11}+15z^{5}a^{11}-20z^{3}a^{11}+21za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-6z^{8}a^{10}+5z^{6}a^{10}+14z^{4}a^{10}-33z^{2}a^{10}-6a^{10}z^{-2}+24a^{10}-z^{9}a^{9}-9z^{7}a^{9}+28z^{5}a^{9}-37z^{3}a^{9}+28za^{9}-14a^{9}z^{-1}+3a^{9}z^{-3}-4z^{8}a^{8}+4z^{6}a^{8}+7z^{4}a^{8}-17z^{2}a^{8}-3a^{8}z^{-2}+13a^{8}-6z^{7}a^{7}+17z^{5}a^{7}-20z^{3}a^{7}+13za^{7}-6a^{7}z^{-1}+a^{7}z^{-3}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

4

-10

4

3

-1

-12

9

4

5

-14

5

7

2

-16

7

6

1

-18

4

8

4

-20

2

4

-2

-22

4

-24

1

2

-1

-26

1

Integral Khovanov Homology

(db, data source)

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