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

Visit L10a70 at Knotilus!

Link Presentations

[edit Notes on L10a70's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X14,6,15,5 X16,12,17,11 X10,4,11,3 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X2,18,3,17
Gauss code {1, -10, 5, -9, 3, -8}, {6, -1, 2, -5, 4, -6, 8, -3, 7, -4, 10, -2, 9, -7}
A Braid Representative
A Morse Link Presentation L10a70 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-3 t(1)^2 t(2)^3+7 t(1) t(2)^3-4 t(2)^3+5 t(1)^2 t(2)^2-9 t(1) t(2)^2+5 t(2)^2-4 t(1)^2 t(2)+7 t(1) t(2)-3 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{9/2}-\frac{4}{q^{9/2}}-5 q^{7/2}+\frac{9}{q^{7/2}}+9 q^{5/2}-\frac{14}{q^{5/2}}-14 q^{3/2}+\frac{17}{q^{3/2}}+\frac{1}{q^{11/2}}+17 \sqrt{q}-\frac{19}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-2 a^3 z^3+z^3 a^{-3} -2 a^3 z- a^{-3} z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +7 a z^3-4 z^3 a^{-1} +4 a z-z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4+4 a^5 z^5-a^5 z^3+9 a^4 z^6+z^6 a^{-4} -8 a^4 z^4-z^4 a^{-4} +3 a^4 z^2+13 a^3 z^7+5 z^7 a^{-3} -19 a^3 z^5-11 z^5 a^{-3} +12 a^3 z^3+5 z^3 a^{-3} -3 a^3 z+2 z a^{-3} - a^{-3} z^{-1} +11 a^2 z^8+8 z^8 a^{-2} -13 a^2 z^6-19 z^6 a^{-2} +a^2 z^4+11 z^4 a^{-2} +a^2 z^2-z^2 a^{-2} + a^{-2} +4 a z^9+4 z^9 a^{-1} +13 a z^7+5 z^7 a^{-1} -42 a z^5-30 z^5 a^{-1} +27 a z^3+19 z^3 a^{-1} -5 a z- a^{-1} z^{-1} +19 z^8-42 z^6+22 z^4-3 z^2 (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         4 4
6        51 -4
4       94  5
2      96   -3
0     108    2
-2    810     2
-4   69      -3
-6  38       5
-8 16        -5
-10 3         3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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