From Knot Atlas
Jump to: navigation, search






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

Visit L11n324 at Knotilus!

Link Presentations

[edit Notes on L11n324's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X17,1,18,4 X21,11,22,10
Gauss code {1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, 4, -5, -10, 6, -8, 7, -11, 8}
A Braid Representative
A Morse Link Presentation L11n324 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(3)^2 t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-2 t(1) t(3)^2 t(2)^2+2 t(3)^2 t(2)^2+t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+2 t(1) t(3)^2 t(2)-t(3)^2 t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1) t(3)^2+t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial - q^{-8} +2 q^{-7} -3 q^{-6} +6 q^{-5} -7 q^{-4} +8 q^{-3} -6 q^{-2} +2 q+6 q^{-1} -3 (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+5 z^4 a^4+10 z^2 a^4+a^4 z^{-2} +7 a^4-3 z^4 a^2-10 z^2 a^2-2 a^2 z^{-2} -9 a^2+2 z^2+ z^{-2} +4 (db)
Kauffman polynomial a^9 z^5-3 a^9 z^3+a^9 z+2 a^8 z^6-6 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-4 a^7 z^5+a^7 z+2 a^6 z^8-5 a^6 z^6+4 a^6 z^4+a^6 z^2+a^5 z^9-a^5 z^7-2 a^5 z^5+8 a^5 z^3-3 a^5 z+4 a^4 z^8-16 a^4 z^6+32 a^4 z^4-26 a^4 z^2-a^4 z^{-2} +9 a^4+a^3 z^9-2 a^3 z^7+2 a^3 z^5+6 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +2 a^2 z^8-9 a^2 z^6+25 a^2 z^4-31 a^2 z^2-2 a^2 z^{-2} +13 a^2+a z^7-a z^5+a z^3-5 a z+2 a z^{-1} +3 z^4-8 z^2- z^{-2} +6 (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       52 3
-3      44  0
-5     42   2
-7    34    1
-9   34     -1
-11  14      3
-13 12       -1
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{5}
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.