From Knot Atlas
Jump to: navigation, search






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

Visit L11n208 at Knotilus!

Link Presentations

[edit Notes on L11n208's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code {1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, 10, 4, -11, 8, 5, -10, -6, 11, -7, -5}
A Braid Representative
A Morse Link Presentation L11n208 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-u^3 v^2+u^2 v^2+u v-v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{1}{q^{9/2}}+q^{7/2}-\frac{1}{q^{7/2}}-q^{5/2}+\frac{1}{q^{5/2}}+q^{3/2}-\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}-2 \sqrt{q}+\frac{1}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^3 z^5+5 a^3 z^3+6 a^3 z+2 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -16 a z^3+5 z^3 a^{-1} -14 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8+7 a z^7+7 z^7 a^{-1} +6 a^2 z^6+7 z^6 a^{-2} +13 z^6-2 a^3 z^5-18 a z^5-16 z^5 a^{-1} -a^4 z^4-12 a^2 z^4-15 z^4 a^{-2} -26 z^4-a^5 z^3+8 a^3 z^3+24 a z^3+15 z^3 a^{-1} -a^6 z^2+2 a^4 z^2+11 a^2 z^2+10 z^2 a^{-2} +18 z^2-a^7 z+a^5 z-7 a^3 z-16 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} (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         1-1
6          0
4       11 0
2      1   1
0      1   1
-2    21    1
-4    1     1
-6  11      0
-8          0
-1011        0
-121         1
Integral Khovanov Homology

(db, data source)

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