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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 L11a41's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X22,15,5,16 X16,7,17,8 X20,18,21,17 X14,10,15,9 X12,19,13,20 X18,13,19,14 X8,21,9,22 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, 6, -2, 11, -7, 8, -6, 3, -4, 5, -8, 7, -5, 9, -3}
A Braid Representative
A Morse Link Presentation L11a41 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{15}{q^{9/2}}-q^{7/2}+\frac{21}{q^{7/2}}+5 q^{5/2}-\frac{26}{q^{5/2}}-12 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{8}{q^{11/2}}+18 \sqrt{q}-\frac{24}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -3 a^3 z^5-7 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +5 a z^3-z^3 a^{-1} +4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-7 a^7 z^5+6 a^7 z^3-3 a^7 z+a^7 z^{-1} +5 a^6 z^8-8 a^6 z^6+2 a^6 z^4+3 a^6 z^2-2 a^6+5 a^5 z^9-a^5 z^7-15 a^5 z^5+21 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-40 a^4 z^6+32 a^4 z^4-8 a^4 z^2+14 a^3 z^9-12 a^3 z^7-23 a^3 z^5+z^5 a^{-3} +33 a^3 z^3-16 a^3 z+3 a^3 z^{-1} +2 a^2 z^{10}+25 a^2 z^8-58 a^2 z^6+5 z^6 a^{-2} +39 a^2 z^4-3 z^4 a^{-2} -11 a^2 z^2+2 a^2+9 a z^9+4 a z^7+12 z^7 a^{-1} -32 a z^5-16 z^5 a^{-1} +24 a z^3+6 z^3 a^{-1} -9 a z-2 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +15 z^8-22 z^6+9 z^4-3 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         81 7
2        104  -6
0       148   6
-2      1412    -2
-4     1212     0
-6    914      5
-8   612       -6
-10  29        7
-12 16         -5
-14 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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