L10a108

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	[edit L10a108 Quick Notes] Two interlinked trefoil knots (3_1).

[edit L10a108 Further Notes and Views]

An indefinitely extended pattern made up of mutually-interlinked trefoils (i.e. overlapping L10a108 links).

Symmetrical trefoils.

Simple square depiction.

Link Presentations

[edit Notes on L10a108's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_20,17,11,18 X_18,13,19,14 X_14,19,15,20 X_2,11,3,12 X_8,15,9,16
Gauss code	{1, -9, 4, -5, 3, -4, 2, -10, 5, -3}, {9, -1, 7, -8, 10, -2, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-t(1)^{3}-2t(2)^{3}t(1)^{2}+6t(2)^{2}t(1)^{2}-6t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-6t(2)^{2}t(1)+6t(2)t(1)-2t(1)-t(2)^{3}+2t(2)^{2}-2t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {14}{q^{13/2}}}+{\frac {14}{q^{15/2}}}-{\frac {13}{q^{17/2}}}+{\frac {11}{q^{19/2}}}-{\frac {6}{q^{21/2}}}+{\frac {3}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-1}-4za^{11}-5a^{11}z^{-1}+6z^{3}a^{9}+14za^{9}+8a^{9}z^{-1}-3z^{5}a^{7}-10z^{3}a^{7}-11za^{7}-4a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+2z^{3}a^{15}-za^{15}-3z^{6}a^{14}+6z^{4}a^{14}-5z^{2}a^{14}+2a^{14}-4z^{7}a^{13}+4z^{5}a^{13}+z^{3}a^{13}-za^{13}-a^{13}z^{-1}-3z^{8}a^{12}-5z^{6}a^{12}+21z^{4}a^{12}-22z^{2}a^{12}+9a^{12}-z^{9}a^{11}-11z^{7}a^{11}+23z^{5}a^{11}-16z^{3}a^{11}+9za^{11}-5a^{11}z^{-1}-7z^{8}a^{10}-z^{6}a^{10}+29z^{4}a^{10}-35z^{2}a^{10}+14a^{10}-z^{9}a^{9}-13z^{7}a^{9}+32z^{5}a^{9}-32z^{3}a^{9}+22za^{9}-8a^{9}z^{-1}-4z^{8}a^{8}-2z^{6}a^{8}+18z^{4}a^{8}-19z^{2}a^{8}+8a^{8}-6z^{7}a^{7}+13z^{5}a^{7}-15z^{3}a^{7}+12za^{7}-4a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

5

-10

5

3

-2

-12

9

5

4

-14

6

0

-16

7

8

-1

-18

4

6

2

-20

2

7

-5

-22

1

4

3

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

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