From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n4 at Knotilus!

Link Presentations

[edit Notes on L10n4's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X15,2,16,3
Gauss code {1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, -10, -2, 3, 8, -9, 7}
A Braid Representative
A Morse Link Presentation L10n4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-2 u v^4+2 u v^3-2 u v^2-2 v^3+2 v^2-2 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{7/2}+2 q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+6 a z^5-z^5 a^{-1} -5 a^3 z^3+12 a z^3-4 z^3 a^{-1} +a^5 z-8 a^3 z+9 a z-4 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial 3 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6-2 a^4 z^4+3 a^4 z^2-2 a^4+2 a^3 z^7-9 a^3 z^5+z^5 a^{-3} +20 a^3 z^3-3 z^3 a^{-3} -16 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^8-2 a^2 z^6+2 z^6 a^{-2} -6 z^4 a^{-2} +7 a^2 z^2+3 z^2 a^{-2} -3 a^2- a^{-2} +4 a z^7+2 z^7 a^{-1} -16 a z^5-6 z^5 a^{-1} +25 a z^3+5 z^3 a^{-1} -13 a z-3 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +z^8-z^6-4 z^4+7 z^2-3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
8        11
6       1 -1
4      21 1
2     21  -1
0    32   1
-2   33    0
-4  12     -1
-6 13      2
-811       0
-102        2
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-4 {\mathbb Z}^{2} {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}_2 {\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.