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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 L11n66's Link Presentations]

Planar diagram presentation X6172 X3,10,4,11 X7,14,8,15 X15,22,16,5 X9,17,10,16 X21,9,22,8 X17,21,18,20 X13,18,14,19 X19,12,20,13 X2536 X11,4,12,1
Gauss code {1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}
A Braid Representative
A Morse Link Presentation L11n66 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^5-5 u v^4+8 u v^3-4 u v^2+u v+v^4-4 v^3+8 v^2-5 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial \frac{12}{q^{9/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{9}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-1} -a^7 z^3-4 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+7 a^5 z^3+8 a^5 z+4 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3 (db)
Kauffman polynomial 3 a^9 z^3-4 a^9 z+a^9 z^{-1} +a^8 z^6+4 a^8 z^4-4 a^8 z^2+a^8+5 a^7 z^7-13 a^7 z^5+25 a^7 z^3-16 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-16 a^6 z^6+23 a^6 z^4-12 a^6 z^2+3 a^6+2 a^5 z^9+8 a^5 z^7-37 a^5 z^5+49 a^5 z^3-25 a^5 z+4 a^5 z^{-1} +11 a^4 z^8-29 a^4 z^6+23 a^4 z^4-9 a^4 z^2+2 a^4+2 a^3 z^9+7 a^3 z^7-35 a^3 z^5+35 a^3 z^3-15 a^3 z+2 a^3 z^{-1} +5 a^2 z^8-11 a^2 z^6+2 a^2 z^4+a^2+4 a z^7-11 a z^5+8 a z^3-2 a z+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       41 -3
-2      73  4
-4     65   -1
-6    76    1
-8   56     1
-10  47      -3
-12 26       4
-14 3        -3
-162         2
Integral Khovanov Homology

(db, data source)

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

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