From Knot Atlas
Jump to: navigation, search






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

Visit L11a281 at Knotilus!

Link Presentations

[edit Notes on L11a281's Link Presentations]

Planar diagram presentation X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X4,22,5,21 X14,5,15,6 X20,13,21,14 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code {1, -5, 3, -6, 7, -10, 11, -2}, {2, -1, 9, -3, 8, -7, 4, -9, 5, -11, 10, -8, 6, -4}
A Braid Representative
A Morse Link Presentation L11a281 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-5 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-3 t(1)^3 t(2)^3+10 t(1)^2 t(2)^3-11 t(1) t(2)^3+3 t(2)^3+3 t(1)^3 t(2)^2-11 t(1)^2 t(2)^2+10 t(1) t(2)^2-3 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-5 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial -\frac{18}{q^{9/2}}-q^{7/2}+\frac{23}{q^{7/2}}+4 q^{5/2}-\frac{27}{q^{5/2}}-9 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{11}{q^{11/2}}+16 \sqrt{q}-\frac{23}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^5 z^5-a^5 z^3+a^5 z+a^5 z^{-1} +a^3 z^7+2 a^3 z^5-a^3 z^3-4 a^3 z-a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} +2 a z-z a^{-1} (db)
Kauffman polynomial -4 a^4 z^{10}-4 a^2 z^{10}-10 a^5 z^9-20 a^3 z^9-10 a z^9-10 a^6 z^8-11 a^4 z^8-12 a^2 z^8-11 z^8-5 a^7 z^7+17 a^5 z^7+43 a^3 z^7+13 a z^7-8 z^7 a^{-1} -a^8 z^6+21 a^6 z^6+40 a^4 z^6+39 a^2 z^6-4 z^6 a^{-2} +17 z^6+9 a^7 z^5-4 a^5 z^5-30 a^3 z^5-5 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +a^8 z^4-10 a^6 z^4-28 a^4 z^4-33 a^2 z^4+5 z^4 a^{-2} -11 z^4-3 a^7 z^3+a^5 z^3+7 a^3 z^3-3 a z^3-5 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2+5 a^4 z^2+7 a^2 z^2-z^2 a^{-2} +2 z^2-a^7 z+a^5 z+2 a^3 z+a z+z a^{-1} +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 \
8           11
6          3 -3
4         61 5
2        103  -7
0       136   7
-2      1411    -3
-4     1312     1
-6    1014      4
-8   813       -5
-10  411        7
-12 17         -6
-14 4          4
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.