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

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Link Presentations

[edit Notes on L10a67's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X20,16,7,15 X16,20,17,19 X12,17,13,18 X6718 X4,13,5,14
Gauss code {1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -8, 10, -4, 6, -7, 8, -5, 7, -6}
A Braid Representative
A Morse Link Presentation L10a67 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+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-4 t(2)+1}{t(1) t(2)^2} (db)
Jones polynomial -\frac{7}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{7}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial z^3 a^9+3 z a^9+a^9 z^{-1} -2 z^5 a^7-8 z^3 a^7-8 z a^7-2 a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+6 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} (db)
Kauffman polynomial a^{13} z^3-a^{13} z+2 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-2 a^{11} z^3+a^{11} z+4 a^{10} z^6-6 a^{10} z^4+4 a^{10} z^2+4 a^9 z^7-8 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-6 a^8 z^6+a^8 z^4+a^7 z^9+3 a^7 z^7-20 a^7 z^5+24 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +5 a^6 z^8-19 a^6 z^6+20 a^6 z^4-8 a^6 z^2+a^6+a^5 z^9-14 a^5 z^5+23 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4-3 a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-5 a^3 z+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 \
0          11
-2         1 -1
-4        31 2
-6       32  -1
-8      42   2
-10     33    0
-12    44     0
-14   23      1
-16  24       -2
-18 13        2
-20 1         -1
-221          1
Integral Khovanov Homology

(db, data source)

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