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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 L11a143's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X20,13,21,14 X14,21,15,22 X2,18,3,17
Gauss code {1, -11, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 9, -10, 7, -8, 11, -2, 4, -9, 10, -4}
A Braid Representative
A Morse Link Presentation L11a143 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-6 t(1)^2 t(2)^3+11 t(1) t(2)^3-4 t(2)^3+7 t(1)^2 t(2)^2-15 t(1) t(2)^2+7 t(2)^2-4 t(1)^2 t(2)+11 t(1) t(2)-6 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial \frac{27}{q^{9/2}}-\frac{27}{q^{7/2}}+\frac{22}{q^{5/2}}+q^{3/2}-\frac{17}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{11}{q^{15/2}}+\frac{18}{q^{13/2}}-\frac{24}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+a^7 z^3-a^7 z-a^7 z^{-1} -a^5 z^7-2 a^5 z^5+a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-5 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3+a z (db)
Kauffman polynomial -z^5 a^{11}-5 z^6 a^{10}+4 z^4 a^{10}-11 z^7 a^9+14 z^5 a^9-4 z^3 a^9-14 z^8 a^8+19 z^6 a^8-5 z^4 a^8-2 z^2 a^8+a^8-10 z^9 a^7+3 z^7 a^7+17 z^5 a^7-12 z^3 a^7+3 z a^7-a^7 z^{-1} -3 z^{10} a^6-20 z^8 a^6+47 z^6 a^6-24 z^4 a^6-3 z^2 a^6+3 a^6-17 z^9 a^5+24 z^7 a^5+4 z^5 a^5-17 z^3 a^5+10 z a^5-3 a^5 z^{-1} -3 z^{10} a^4-13 z^8 a^4+37 z^6 a^4-23 z^4 a^4+3 a^4-7 z^9 a^3+6 z^7 a^3+11 z^5 a^3-16 z^3 a^3+9 z a^3-2 a^3 z^{-1} -7 z^8 a^2+13 z^6 a^2-6 z^4 a^2-4 z^7 a+9 z^5 a-7 z^3 a+2 z a-z^6+2 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 \
4           1-1
2          3 3
0         61 -5
-2        113  8
-4       127   -5
-6      1510    5
-8     1313     0
-10    1114      -3
-12   713       6
-14  411        -7
-16 17         6
-18 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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