L11n453

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_19,22,20,15 X_21,14,22,11 X_13,20,14,21 X_9,18,10,19 X_11,10,12,5 X_4,17,1,18
Gauss code	{1, 4, -3, -11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {-4, -5, 11, 9, -6, 8, -7, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uvw^{2}x^{2}-uvw^{2}x-uvwx^{2}-uw^{2}x^{2}+uw^{2}x+uwx^{2}-uwx-vwx+vw+vx-v-w-x+2}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{25/2}}}-{\frac {3}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {6}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}+a^{13}z^{-3}+z^{5}a^{11}+4z^{3}a^{11}+za^{11}-5a^{11}z^{-1}-3a^{11}z^{-3}-z^{7}a^{9}-4z^{5}a^{9}+11za^{9}+10a^{9}z^{-1}+3a^{9}z^{-3}-z^{7}a^{7}-6z^{5}a^{7}-12z^{3}a^{7}-11za^{7}-5a^{7}z^{-1}-a^{7}z^{-3}$ (db)
Kauffman polynomial	$-z^{2}a^{16}-3z^{3}a^{15}+3za^{15}-z^{6}a^{14}+z^{4}a^{14}-z^{2}a^{14}-3z^{7}a^{13}+12z^{5}a^{13}-18z^{3}a^{13}+12za^{13}-5a^{13}z^{-1}+a^{13}z^{-3}-3z^{8}a^{12}+12z^{6}a^{12}-12z^{4}a^{12}-3z^{2}a^{12}-3a^{12}z^{-2}+10a^{12}-z^{9}a^{11}+14z^{5}a^{11}-25z^{3}a^{11}+21za^{11}-12a^{11}z^{-1}+3a^{11}z^{-3}-4z^{8}a^{10}+16z^{6}a^{10}-10z^{4}a^{10}-15z^{2}a^{10}-6a^{10}z^{-2}+19a^{10}-z^{9}a^{9}+2z^{7}a^{9}+8z^{5}a^{9}-22z^{3}a^{9}+23za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-z^{8}a^{8}+3z^{6}a^{8}+3z^{4}a^{8}-12z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{7}a^{7}+6z^{5}a^{7}-12z^{3}a^{7}+11za^{7}-5a^{7}z^{-1}+a^{7}z^{-3}$ (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

-6

1

-8

1

0

-10

3

-12

1

-14

6

3

-16

2

-18

6

4

2

-20

1

2

4

3

-22

2

0

-24

2

-26

1

-1

Integral Khovanov Homology

(db, data source)

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