From Knot Atlas
Jump to: navigation, search






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

Visit L11n280 at Knotilus!

Link Presentations

[edit Notes on L11n280's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code {1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {-8, 2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}
A Braid Representative
A Morse Link Presentation L11n280 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^3+2 q^2-5 q+8-7 q^{-1} +9 q^{-2} -6 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-4 a^2 z^4+2 z^4+2 a^4 z^2-7 a^2 z^2-z^2 a^{-2} +6 z^2+2 a^4-8 a^2-2 a^{-2} +8+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+5 a^5 z^3+3 a^4 z^8-7 a^4 z^6-a^4 z^2-2 a^4 z^{-2} +4 a^4+a^3 z^9+3 a^3 z^7-17 a^3 z^5+18 a^3 z^3+z^3 a^{-3} -12 a^3 z-2 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +5 a^2 z^8-15 a^2 z^6+15 a^2 z^4+2 z^4 a^{-2} -11 a^2 z^2-2 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +10 a^2+ a^{-2} +a z^9+a z^7+z^7 a^{-1} -11 a z^5-3 z^5 a^{-1} +23 a z^3+11 z^3 a^{-1} -21 a z-11 z a^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} +2 z^8-7 z^6+14 z^4-9 z^2-4 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 \
7         1-1
5        1 1
3       41 -3
1      41  3
-1     56   1
-3    431   2
-5   25     3
-7  44      0
-9 14       3
-11 2        -2
-131         1
Integral Khovanov Homology

(db, data source)

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