From Knot Atlas
Jump to: navigation, search






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

Visit L11a334 at Knotilus!

Link Presentations

[edit Notes on L11a334's Link Presentations]

Planar diagram presentation X10,1,11,2 X20,6,21,5 X14,3,15,4 X4,15,5,16 X22,20,9,19 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X6,22,7,21
Gauss code {1, -9, 3, -4, 2, -11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, 5, -2, 11, -5}
A Braid Representative
A Morse Link Presentation L11a334 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-4 t(2)^3 t(1)^2+11 t(2)^2 t(1)^2-11 t(2) t(1)^2+3 t(1)^2+3 t(2)^3 t(1)-11 t(2)^2 t(1)+11 t(2) t(1)-4 t(1)+2 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{8}{q^{9/2}}-4 q^{7/2}+\frac{14}{q^{7/2}}+9 q^{5/2}-\frac{20}{q^{5/2}}-15 q^{3/2}+\frac{22}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+20 \sqrt{q}-\frac{23}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+4 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-5 a^3 z+2 a z-2 z a^{-1} +z a^{-3} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-6 a^3 z^9-12 a z^9-6 z^9 a^{-1} -8 a^4 z^8-14 a^2 z^8-7 z^8 a^{-2} -13 z^8-6 a^5 z^7+15 a z^7+5 z^7 a^{-1} -4 z^7 a^{-3} -3 a^6 z^6+12 a^4 z^6+35 a^2 z^6+15 z^6 a^{-2} -z^6 a^{-4} +36 z^6-a^7 z^5+9 a^5 z^5+16 a^3 z^5+7 a z^5+10 z^5 a^{-1} +9 z^5 a^{-3} +4 a^6 z^4-10 a^4 z^4-27 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -24 z^4+2 a^7 z^3-8 a^5 z^3-21 a^3 z^3-13 a z^3-8 z^3 a^{-1} -6 z^3 a^{-3} -a^6 z^2+2 a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +6 z^2-a^7 z+5 a^5 z+9 a^3 z+4 a z+2 z a^{-1} +z a^{-3} +a^4-a^5 z^{-1} -a^3 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 \
10           1-1
8          3 3
6         61 -5
4        93  6
2       116   -5
0      129    3
-2     1112     1
-4    911      -2
-6   511       6
-8  39        -6
-10 16         5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.