L11n67

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_9,20,10,21 X_13,18,14,19 X_7,14,8,15 X_17,8,18,9 X_19,12,20,13 X_15,22,16,5 X_21,16,22,17 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -3, 2, -11, 7, -4, 5, -8, 9, -6, 4, -7, 3, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {5t(1)t(2)^{3}-t(2)^{3}-9t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-9t(2)-t(1)+5}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {14}{q^{13/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{17/2}}}+{\frac {10}{q^{19/2}}}-{\frac {6}{q^{21/2}}}+{\frac {3}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+z^{3}a^{11}+2za^{11}+a^{11}z^{-1}-z^{5}a^{9}+4za^{9}+2a^{9}z^{-1}-3z^{5}a^{7}-9z^{3}a^{7}-7za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-6z^{4}a^{14}+11z^{2}a^{14}-4a^{14}-3z^{7}a^{13}+3z^{5}a^{13}-z^{3}a^{13}-za^{13}+a^{13}z^{-1}-5z^{8}a^{12}+13z^{6}a^{12}-27z^{4}a^{12}+28z^{2}a^{12}-9a^{12}-2z^{9}a^{11}-5z^{7}a^{11}+14z^{5}a^{11}-11z^{3}a^{11}+za^{11}+a^{11}z^{-1}-10z^{8}a^{10}+19z^{6}a^{10}-17z^{4}a^{10}+11z^{2}a^{10}-4a^{10}-2z^{9}a^{9}-8z^{7}a^{9}+24z^{5}a^{9}-24z^{3}a^{9}+12za^{9}-2a^{9}z^{-1}-5z^{8}a^{8}+3z^{6}a^{8}+8z^{4}a^{8}-7z^{2}a^{8}+2a^{8}-6z^{7}a^{7}+12z^{5}a^{7}-12z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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

5

-10

6

3

-3

-12

8

5

3

-14

7

0

-16

7

0

-18

3

7

4

-20

3

7

-4

-22

3

-24

3

-3

Integral Khovanov Homology

(db, data source)

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