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
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   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)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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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