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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 L11a67's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X14,9,15,10 X10,20,11,19 X8,21,9,22 X18,14,19,13 X20,15,21,16 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -7, 5, -6, 11, -2, 8, -5, 9, -3, 4, -8, 6, -9, 7, -4}
A Braid Representative
A Morse Link Presentation L11a67 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+9 t(2)^4+12 t(1) t(2)^3-16 t(2)^3-16 t(1) t(2)^2+12 t(2)^2+9 t(1) t(2)-5 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{18}{q^{9/2}}-q^{7/2}+\frac{24}{q^{7/2}}+5 q^{5/2}-\frac{29}{q^{5/2}}-13 q^{3/2}+\frac{30}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+20 \sqrt{q}-\frac{26}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} -3 z^5 a^3-6 z^3 a^3-7 z a^3-4 a^3 z^{-1} +z^7 a+3 z^5 a+6 z^3 a+5 z a+3 a z^{-1} -z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -3 a^4 z^{10}-3 a^2 z^{10}-7 a^5 z^9-19 a^3 z^9-12 a z^9-7 a^6 z^8-18 a^4 z^8-29 a^2 z^8-18 z^8-4 a^7 z^7+2 a^5 z^7+20 a^3 z^7+a z^7-13 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+50 a^4 z^6+70 a^2 z^6-5 z^6 a^{-2} +28 z^6+9 a^7 z^5+22 a^5 z^5+27 a^3 z^5+31 a z^5+16 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-6 a^6 z^4-37 a^4 z^4-45 a^2 z^4+2 z^4 a^{-2} -14 z^4-8 a^7 z^3-31 a^5 z^3-45 a^3 z^3-29 a z^3-7 z^3 a^{-1} -a^8 z^2+a^6 z^2+9 a^4 z^2+11 a^2 z^2+4 z^2+3 a^7 z+16 a^5 z+24 a^3 z+15 a z+4 z a^{-1} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1} (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         91 8
2        114  -7
0       159   6
-2      1612    -4
-4     1314     -1
-6    1116      5
-8   713       -6
-10  312        9
-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}^{12}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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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