From Knot Atlas
Jump to: navigation, search






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

Visit L10a92 at Knotilus!

Link Presentations

[edit Notes on L10a92's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X18,14,19,13 X14,5,15,6 X4,17,5,18 X16,7,17,8 X6,15,7,16 X8,20,1,19
Gauss code {1, -4, 2, -7, 6, -9, 8, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 7, -5, 10, -3}
A Braid Representative
A Morse Link Presentation L10a92 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^3-2 u^3 v^2+u^3 v-2 u^2 v^3+5 u^2 v^2-5 u^2 v+u^2+u v^3-5 u v^2+5 u v-2 u+v^2-2 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+2 a^5 z+a^5 z^{-1} -2 a^3 z^5-7 a^3 z^3-7 a^3 z-a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +9 a z^3-3 z^3 a^{-1} +5 a z-2 z a^{-1} (db)
Kauffman polynomial a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+2 a^6 z^2+4 a^5 z^7-7 a^5 z^5+3 a^5 z^3-3 a^5 z+a^5 z^{-1} +3 a^4 z^8-2 a^4 z^6-3 a^4 z^4+2 a^4 z^2-a^4+a^3 z^9+6 a^3 z^7-17 a^3 z^5+z^5 a^{-3} +18 a^3 z^3-2 z^3 a^{-3} -8 a^3 z+a^3 z^{-1} +6 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +8 a^2 z^4-7 z^4 a^{-2} +3 z^2 a^{-2} +a z^9+6 a z^7+4 z^7 a^{-1} -19 a z^5-9 z^5 a^{-1} +22 a z^3+7 z^3 a^{-1} -7 a z-3 z a^{-1} +3 z^8-3 z^6-2 z^4+3 z^2 (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 \
8          11
6         2 -2
4        31 2
2       52  -3
0      63   3
-2     66    0
-4    55     0
-6   36      3
-8  35       -2
-10 14        3
-12 2         -2
-141          1
Integral Khovanov Homology

(db, data source)

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