From Knot Atlas
Jump to: navigation, search






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

Visit L11a1 at Knotilus!

Link L11a1.
A graph, link L11a1.
A part of a link and a part of a graph.

Link Presentations

[edit Notes on L11a1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X20,12,21,11 X22,17,5,18 X18,21,19,22 X12,20,13,19 X16,10,17,9 X2,14,3,13
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, 9, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L11a1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a z+2 z a^{-3} -z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-7 a z^9-13 z^9 a^{-1} -6 z^9 a^{-3} -9 a^2 z^8-17 z^8 a^{-2} -9 z^8 a^{-4} -17 z^8-5 a^3 z^7+5 a z^7+12 z^7 a^{-1} -6 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+20 a^2 z^6+36 z^6 a^{-2} +9 z^6 a^{-4} -4 z^6 a^{-6} +44 z^6+10 a^3 z^5+14 a z^5+16 z^5 a^{-1} +25 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +a^4 z^4-11 a^2 z^4-19 z^4 a^{-2} -z^4 a^{-4} +5 z^4 a^{-6} -25 z^4-4 a^3 z^3-9 a z^3-14 z^3 a^{-1} -18 z^3 a^{-3} -8 z^3 a^{-5} +z^3 a^{-7} +2 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +z^2-a^3 z-4 a z-2 z a^{-1} +3 z a^{-3} +2 z a^{-5} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} 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 \
14           11
12          3 -3
10         61 5
8        93  -6
6       126   6
4      129    -3
2     1312     1
0    1014      4
-2   611       -5
-4  410        6
-6 16         -5
-8 4          4
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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.