L10a108

L10a107

L10a109

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a108 at Knotilus!
	Two interlinked trefoil knots (3_1).

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	X12,1,13,2 X16,7,17,8 X10,5,1,6 X6374 X4,9,5,10 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X8,15,9,16
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 1 -6                   3 1 -2 -8                 5     5 -10               5 3     -2 -12             9 5       4 -14           6 6         0 -16         7 8           -1 -18       4 6             2 -20     2 7               -5 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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