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

Visit L10a78 at Knotilus!

Link Presentations

[edit Notes on L10a78's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X18,12,19,11 X16,5,17,6 X12,18,13,17 X20,16,7,15 X4,19,5,20
Gauss code {1, -2, 3, -10, 7, -5}, {4, -1, 2, -3, 6, -8, 5, -4, 9, -7, 8, -6, 10, -9}
A Braid Representative
A Morse Link Presentation L10a78 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-4 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-4 t(2)+1}{t(1) t(2)^2} (db)
Jones polynomial \frac{8}{q^{9/2}}-\frac{11}{q^{7/2}}-q^{5/2}+\frac{10}{q^{5/2}}+3 q^{3/2}-\frac{10}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{11/2}}-5 \sqrt{q}+\frac{8}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+3 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-10 a^3 z^3-10 a^3 z-3 a^3 z^{-1} +2 a z^5+7 a z^3-z^3 a^{-1} +6 a z-2 z a^{-1} +a z^{-1} (db)
Kauffman polynomial -z^3 a^9-3 z^4 a^8-6 z^5 a^7+5 z^3 a^7-2 z a^7-8 z^6 a^6+10 z^4 a^6-2 z^2 a^6-9 z^7 a^5+19 z^5 a^5-13 z^3 a^5+7 z a^5-2 a^5 z^{-1} -6 z^8 a^4+10 z^6 a^4+4 z^4 a^4-7 z^2 a^4+3 a^4-2 z^9 a^3-5 z^7 a^3+31 z^5 a^3-31 z^3 a^3+12 z a^3-3 a^3 z^{-1} -9 z^8 a^2+31 z^6 a^2-26 z^4 a^2+z^2 a^2+3 a^2-2 z^9 a+3 z^7 a+10 z^5 a-17 z^3 a+5 z a-a z^{-1} -3 z^8+13 z^6-17 z^4+6 z^2+1-z^7 a^{-1} +4 z^5 a^{-1} -5 z^3 a^{-1} +2 z a^{-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 \
6          11
4         2 -2
2        31 2
0       52  -3
-2      53   2
-4     66    0
-6    54     1
-8   36      3
-10  35       -2
-12  3        3
-1413         -2
-161          1
Integral Khovanov Homology

(db, data source)

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

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