From Knot Atlas
Jump to: navigation, search






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

Visit L11a44 at Knotilus!

Link Presentations

[edit Notes on L11a44's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X20,8,21,7 X22,15,5,16 X16,21,17,22 X14,10,15,9 X18,13,19,14 X12,19,13,20 X8,18,9,17 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 6, -2, 11, -8, 7, -6, 4, -5, 9, -7, 8, -3, 5, -4}
A Braid Representative
A Morse Link Presentation L11a44 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+8 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+20 q^{3/2}-24 \sqrt{q}+\frac{23}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-5 a^3 z+9 a z-8 z a^{-1} +3 z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-4 a^3 z^9-13 a z^9-9 z^9 a^{-1} -3 a^4 z^8-9 a^2 z^8-17 z^8 a^{-2} -23 z^8-a^5 z^7+7 a^3 z^7+20 a z^7-5 z^7 a^{-1} -17 z^7 a^{-3} +10 a^4 z^6+41 a^2 z^6+26 z^6 a^{-2} -10 z^6 a^{-4} +67 z^6+4 a^5 z^5+8 a^3 z^5+24 a z^5+50 z^5 a^{-1} +26 z^5 a^{-3} -4 z^5 a^{-5} -12 a^4 z^4-42 a^2 z^4-8 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -46 z^4-6 a^5 z^3-24 a^3 z^3-49 a z^3-49 z^3 a^{-1} -18 z^3 a^{-3} +6 a^4 z^2+14 a^2 z^2-z^2 a^{-2} -3 z^2 a^{-4} +10 z^2+4 a^5 z+16 a^3 z+26 a z+21 z a^{-1} +7 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 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 \
12           1-1
10          3 3
8         71 -6
6        103  7
4       107   -3
2      1410    4
0     1112     1
-2    812      -4
-4   611       5
-6  28        -6
-8 16         5
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
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}^{2}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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.