L11a374

L11a373

L11a375

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a374 at Knotilus!

Link Presentations

[edit Notes on L11a374's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X18,10,19,9 X20,18,21,17 X10,20,1,19 X8,11,9,12 X4,15,5,16 X6,21,7,22
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9, 6, -8}, {9, -1, 2, -3, 10, -4, 7, -6, 8, -7, 11, -5}

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{(u v-u+1) (u v-v+1) \left(u^2 v^2+u v+1\right)}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^7+6 a^7 z^5+12 a^7 z^3+9 a^7 z+a^7 z^{-1} -a^5 z^9-8 a^5 z^7-24 a^5 z^5-33 a^5 z^3-18 a^5 z-a^5 z^{-1} +a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{13}+z a^{13}-2 z^4 a^{12}+z^2 a^{12}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}-4 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-4 z^7 a^9+7 z^5 a^9-2 z^3 a^9-4 z^8 a^8+11 z^6 a^8-9 z^4 a^8+5 z^2 a^8-3 z^9 a^7+10 z^7 a^7-12 z^5 a^7+15 z^3 a^7-9 z a^7+a^7 z^{-1} -z^{10} a^6-z^8 a^6+18 z^6 a^6-29 z^4 a^6+16 z^2 a^6-a^6-5 z^9 a^5+26 z^7 a^5-46 z^5 a^5+40 z^3 a^5-18 z a^5+a^5 z^{-1} -z^{10} a^4+2 z^8 a^4+9 z^6 a^4-23 z^4 a^4+12 z^2 a^4-2 z^9 a^3+12 z^7 a^3-24 z^5 a^3+20 z^3 a^3-7 z a^3-z^8 a^2+6 z^6 a^2-11 z^4 a^2+6 z^2 a^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 4 1     3 -6               3 3       0 -8             5 3         2 -10           3 3           0 -12         4 5             -1 -14       2 3               1 -16     2 4                 -2 -18   1 3                   2 -20   1                     -1 -22 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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.

L11a373 L11a375