From Knot Atlas
Jump to: navigation, search






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

Visit L11a280 at Knotilus!

Link Presentations

[edit Notes on L11a280's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X22,13,9,14 X14,6,15,5 X4,22,5,21 X18,8,19,7 X20,16,21,15 X16,20,17,19 X6,18,7,17
Gauss code {1, -2, 3, -7, 6, -11, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -10, 11, -8, 10, -9, 7, -5}
A Braid Representative
A Morse Link Presentation L11a280 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{11/2}+3 q^{9/2}-5 q^{7/2}+8 q^{5/2}-12 q^{3/2}+13 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -2 a^3 z-z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +5 a z^3+4 z^3 a^{-1} +3 a z+a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-4 z^5 a^{-5} -3 a^5 z^3+4 z^3 a^{-5} +a^5 z-z a^{-5} +3 z^8 a^{-4} +5 a^4 z^6-13 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +2 a^4 z^2-5 z^2 a^{-4} +3 z^9 a^{-3} +6 a^3 z^7-10 z^7 a^{-3} -7 a^3 z^5+7 z^5 a^{-3} +4 a^3 z^3+z^3 a^{-3} -a^3 z-z a^{-3} +z^{10} a^{-2} +5 a^2 z^8+4 z^8 a^{-2} -4 a^2 z^6-24 z^6 a^{-2} -2 a^2 z^4+28 z^4 a^{-2} +2 a^2 z^2-9 z^2 a^{-2} +3 a z^9+6 z^9 a^{-1} +a z^7-16 z^7 a^{-1} -11 a z^5+10 z^5 a^{-1} +9 a z^3-z^3 a^{-1} -4 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+6 z^8-20 z^6+16 z^4-5 z^2-1 (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           11
10          2 -2
8         31 2
6        52  -3
4       73   4
2      65    -1
0     87     1
-2    68      2
-4   46       -2
-6  26        4
-8 14         -3
-10 2          2
-121           -1
Integral Khovanov Homology

(db, data source)

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