From Knot Atlas
Jump to: navigation, search






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

Visit L11n386 at Knotilus!

Link Presentations

[edit Notes on L11n386's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X2,14,3,13
Gauss code {1, -11, -5, 3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}
A Braid Representative
A Morse Link Presentation L11n386 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-8} +2 q^{-7} -4 q^{-6} +6 q^{-5} -8 q^{-4} +9 q^{-3} -6 q^{-2} +2 q+7 q^{-1} -3 (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^4 a^6-3 z^2 a^6-a^6 z^{-2} -3 a^6+z^6 a^4+5 z^4 a^4+11 z^2 a^4+4 a^4 z^{-2} +10 a^4-3 z^4 a^2-10 z^2 a^2-5 a^2 z^{-2} -11 a^2+2 z^2+2 z^{-2} +4 (db)
Kauffman polynomial a^9 z^5-3 a^9 z^3+a^9 z+2 a^8 z^6-5 a^8 z^4+a^8 z^2+3 a^7 z^7-9 a^7 z^5+9 a^7 z^3-5 a^7 z+a^7 z^{-1} +3 a^6 z^8-10 a^6 z^6+14 a^6 z^4-9 a^6 z^2-a^6 z^{-2} +5 a^6+a^5 z^9+2 a^5 z^7-17 a^5 z^5+35 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-20 a^4 z^6+39 a^4 z^4-34 a^4 z^2-4 a^4 z^{-2} +16 a^4+a^3 z^9-8 a^3 z^5+26 a^3 z^3-26 a^3 z+9 a^3 z^{-1} +2 a^2 z^8-8 a^2 z^6+23 a^2 z^4-32 a^2 z^2-5 a^2 z^{-2} +17 a^2+a z^7-a z^5+3 a z^3-10 a z+5 a z^{-1} +3 z^4-8 z^2-2 z^{-2} +7 (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
1        1 -1
-1       62 4
-3      45  1
-5     52   3
-7    34    1
-9   35     -2
-11  13      2
-13 13       -2
-15 1        1
-171         -1
Integral Khovanov Homology

(db, data source)

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