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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 L11a140's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,19,15,20 X20,13,21,14 X18,22,19,21 X6,15,1,16
Gauss code {1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -6, 9, -8, 11, -5, 7, -10, 8, -9, 10, -3}
A Braid Representative
A Morse Link Presentation L11a140 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(t(1) t(2)^2-t(2)^2-3 t(1) t(2)+2 t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-2 t(1) t(2)+3 t(2)+t(1)-1\right)}{t(1) t(2)^2} (db)
Jones polynomial -q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{24}{\sqrt{q}}+\frac{26}{q^{3/2}}-\frac{26}{q^{5/2}}+\frac{22}{q^{7/2}}-\frac{16}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+3 z^3 a^5+3 z a^5-3 z^5 a^3-6 z^3 a^3-3 z a^3+a^3 z^{-1} +z^7 a+3 z^5 a+4 z^3 a-a z^{-1} -z^5 a^{-1} -z^3 a^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+7 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+9 a^6 z^4-2 a^6 z^2+6 a^5 z^9-a^5 z^7-20 a^5 z^5+21 a^5 z^3-6 a^5 z+2 a^4 z^{10}+18 a^4 z^8-47 a^4 z^6+34 a^4 z^4-8 a^4 z^2+14 a^3 z^9-9 a^3 z^7-29 a^3 z^5+z^5 a^{-3} +29 a^3 z^3-5 a^3 z-a^3 z^{-1} +2 a^2 z^{10}+24 a^2 z^8-53 a^2 z^6+5 z^6 a^{-2} +31 a^2 z^4-4 z^4 a^{-2} -6 a^2 z^2+a^2+8 a z^9+7 a z^7+11 z^7 a^{-1} -34 a z^5-15 z^5 a^{-1} +21 a z^3+6 z^3 a^{-1} -a z-a z^{-1} +13 z^8-16 z^6+4 z^4-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 \
8           11
6          4 -4
4         71 6
2        104  -6
0       147   7
-2      1311    -2
-4     1313     0
-6    1014      4
-8   612       -6
-10  310        7
-12 16         -5
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
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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