From Knot Atlas
Jump to: navigation, search






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

Visit L10n18 at Knotilus!

Link Presentations

[edit Notes on L10n18's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X2,18,3,17
Gauss code {1, -10, 5, -3}, {-4, -1, 2, -5, -7, 8, -9, 4, -6, 7, -8, 6, 10, -2, 3, 9}
A Braid Representative
A Morse Link Presentation L10n18 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+2 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^5-z^5 a^{-1} -a^3 z^3+6 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -3 a^3 z+9 a z-8 z a^{-1} +2 z a^{-3} -a^3 z^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -z^8 a^{-2} -z^8-a^3 z^7-2 a z^7-3 z^7 a^{-1} -2 z^7 a^{-3} -a^4 z^6-a^2 z^6+3 z^6 a^{-2} -z^6 a^{-4} +4 z^6+6 a^3 z^5+14 a z^5+17 z^5 a^{-1} +9 z^5 a^{-3} +5 a^4 z^4+8 a^2 z^4+4 z^4 a^{-2} +4 z^4 a^{-4} +3 z^4-9 a^3 z^3-26 a z^3-26 z^3 a^{-1} -9 z^3 a^{-3} -5 a^4 z^2-12 a^2 z^2-10 z^2 a^{-2} -3 z^2 a^{-4} -14 z^2+6 a^3 z+17 a z+15 z a^{-1} +4 z a^{-3} +a^4+4 a^2+4 a^{-2} + a^{-4} +7-a^3 z^{-1} -4 a z^{-1} -4 a^{-1} z^{-1} - a^{-3} z^{-1} (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 \
10          1-1
8         1 1
6        11 0
4      121  0
2     111   1
0    142    1
-2   112     2
-4   11      0
-6 111       1
-8           0
-101          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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.