From Knot Atlas
Jump to: navigation, search






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

Visit L10n79 at Knotilus!

Link Presentations

[edit Notes on L10n79's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11
Gauss code {1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, {-8, 2, 4, -5, 9, 6, -10, 7}
A Braid Representative
A Morse Link Presentation L10n79 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) (w-1) (v w+1)}{\sqrt{u} v w} (db)
Jones polynomial 2 q^{-2} -2 q^{-3} +5 q^{-4} -5 q^{-5} +6 q^{-6} -5 q^{-7} +4 q^{-8} -2 q^{-9} + q^{-10} (db)
Signature -4 (db)
HOMFLY-PT polynomial z^4 a^8+3 z^2 a^8+a^8 z^{-2} +3 a^8-z^6 a^6-5 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} -9 a^6+2 z^4 a^4+7 z^2 a^4+a^4 z^{-2} +6 a^4 (db)
Kauffman polynomial a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-a^{10} z^4-2 a^{10} z^2+2 a^9 z^7-3 a^9 z^5+3 a^9 z^3+a^8 z^8-a^8 z^6+3 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-12 a^6 z^2-2 a^6 z^{-2} +9 a^6+a^5 z^7-3 a^5 z^5+7 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +6 a^4 (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 \
-3        22
-5       220
-7      3  3
-9     22  0
-11    43   1
-13   12    1
-15  34     -1
-17 13      2
-19 1       -1
-211        1
Integral Khovanov Homology

(db, data source)

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

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.