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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 L10a56's Link Presentations]

Planar diagram presentation X8192 X16,7,17,8 X10,4,11,3 X12,5,13,6 X20,12,7,11 X18,13,19,14 X2,15,3,16 X4,19,5,20 X14,10,15,9 X6,18,1,17
Gauss code {1, -7, 3, -8, 4, -10}, {2, -1, 9, -3, 5, -4, 6, -9, 7, -2, 10, -6, 8, -5}
A Braid Representative
A Morse Link Presentation L10a56 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-4 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-2 u v^4+8 u v^3-13 u v^2+8 u v-2 u+v^4-4 v^3+6 v^2-4 v+1}{u v^2} (db)
Jones polynomial -\frac{11}{q^{9/2}}-q^{7/2}+\frac{16}{q^{7/2}}+5 q^{5/2}-\frac{21}{q^{5/2}}-11 q^{3/2}+\frac{22}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+16 \sqrt{q}-\frac{21}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-3 a^3 z^3+3 a z^3-z^3 a^{-1} -a z+a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -5 a^3 z^9-5 a z^9-12 a^4 z^8-24 a^2 z^8-12 z^8-11 a^5 z^7-14 a^3 z^7-14 a z^7-11 z^7 a^{-1} -5 a^6 z^6+16 a^4 z^6+42 a^2 z^6-5 z^6 a^{-2} +16 z^6-a^7 z^5+16 a^5 z^5+39 a^3 z^5+39 a z^5+16 z^5 a^{-1} -z^5 a^{-3} +4 a^6 z^4-4 a^4 z^4-16 a^2 z^4+4 z^4 a^{-2} -4 z^4-6 a^5 z^3-18 a^3 z^3-18 a z^3-6 z^3 a^{-1} -a^3 z-a z-a^2+a^3 z^{-1} +a 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         4 -4
4        71 6
2       94  -5
0      127   5
-2     1110    -1
-4    1011     -1
-6   712      5
-8  49       -5
-10 17        6
-12 4         -4
-141          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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