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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 L11a227's Link Presentations]

Planar diagram presentation X8192 X20,9,21,10 X4758 X16,5,17,6 X14,18,15,17 X6,15,1,16 X22,14,7,13 X18,22,19,21 X2,11,3,12 X12,3,13,4 X10,19,11,20
Gauss code {1, -9, 10, -3, 4, -6}, {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}
A Braid Representative
A Morse Link Presentation L11a227 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-3 u^2 v-2 u v^4+10 u v^3-15 u v^2+10 u v-2 u-3 v^3+8 v^2-5 v+1}{u v^2} (db)
Jones polynomial \frac{23}{q^{9/2}}-\frac{24}{q^{7/2}}+\frac{21}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{14}{q^{13/2}}-\frac{20}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -3 a^7 z^3-5 a^7 z-2 a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^5+a z^3-a z (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-6 z^7 a^9+9 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -8 z^8 a^8+12 z^6 a^8-11 z^4 a^8+4 z^2 a^8-6 z^9 a^7-z^7 a^7+19 z^5 a^7-26 z^3 a^7+12 z a^7-2 a^7 z^{-1} -2 z^{10} a^6-16 z^8 a^6+42 z^6 a^6-37 z^4 a^6+12 z^2 a^6-a^6-13 z^9 a^5+15 z^7 a^5+15 z^5 a^5-26 z^3 a^5+12 z a^5-2 a^5 z^{-1} -2 z^{10} a^4-17 z^8 a^4+48 z^6 a^4-34 z^4 a^4+8 z^2 a^4-7 z^9 a^3+5 z^7 a^3+16 z^5 a^3-14 z^3 a^3+5 z a^3-a^3 z^{-1} -9 z^8 a^2+20 z^6 a^2-11 z^4 a^2+z^2 a^2-5 z^7 a+10 z^5 a-4 z^3 a-z a-z^6+z^4 (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 \
4           1-1
2          4 4
0         61 -5
-2        104  6
-4       127   -5
-6      129    3
-8     1112     1
-10    912      -3
-12   511       6
-14  39        -6
-16 16         5
-18 2          -2
-201           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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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