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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 L11a374's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X22,7,11,8 X18,10,19,9 X20,18,21,17 X10,20,1,19 X8,11,9,12 X4,15,5,16 X6,21,7,22
Gauss code {1, -2, 3, -10, 4, -11, 5, -9, 6, -8}, {9, -1, 2, -3, 10, -4, 7, -6, 8, -7, 11, -5}
A Braid Representative
A Morse Link Presentation L11a374 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u v-u+1) (u v-v+1) \left(u^2 v^2+u v+1\right)}{u^2 v^2} (db)
Jones polynomial \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^7 z^7+6 a^7 z^5+12 a^7 z^3+9 a^7 z+a^7 z^{-1} -a^5 z^9-8 a^5 z^7-24 a^5 z^5-33 a^5 z^3-18 a^5 z-a^5 z^{-1} +a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z (db)
Kauffman polynomial -z^3 a^{13}+z a^{13}-2 z^4 a^{12}+z^2 a^{12}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}-4 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-4 z^7 a^9+7 z^5 a^9-2 z^3 a^9-4 z^8 a^8+11 z^6 a^8-9 z^4 a^8+5 z^2 a^8-3 z^9 a^7+10 z^7 a^7-12 z^5 a^7+15 z^3 a^7-9 z a^7+a^7 z^{-1} -z^{10} a^6-z^8 a^6+18 z^6 a^6-29 z^4 a^6+16 z^2 a^6-a^6-5 z^9 a^5+26 z^7 a^5-46 z^5 a^5+40 z^3 a^5-18 z a^5+a^5 z^{-1} -z^{10} a^4+2 z^8 a^4+9 z^6 a^4-23 z^4 a^4+12 z^2 a^4-2 z^9 a^3+12 z^7 a^3-24 z^5 a^3+20 z^3 a^3-7 z a^3-z^8 a^2+6 z^6 a^2-11 z^4 a^2+6 z^2 a^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 \
2           1-1
0          1 1
-2         21 -1
-4        41  3
-6       33   0
-8      53    2
-10     33     0
-12    45      -1
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\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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