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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 L11a277's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X8,9,1,10 X4,13,5,14 X20,15,21,16 X18,8,19,7 X6,20,7,19 X22,17,9,18 X16,21,17,22
Gauss code {1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -11, 10, -8, 9, -7, 11, -10}
A Braid Representative
A Morse Link Presentation L11a277 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-t(1)^3 t(2)^4+t(1)^2 t(2)^4-t(1) t(2)^4+t(1)^3 t(2)^3-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^9 \left(-z^7\right)-6 a^9 z^5-11 a^9 z^3-7 a^9 z-a^9 z^{-1} +a^7 z^9+8 a^7 z^7+23 a^7 z^5+30 a^7 z^3+18 a^7 z+3 a^7 z^{-1} -a^5 z^7-7 a^5 z^5-16 a^5 z^3-13 a^5 z-2 a^5 z^{-1} (db)
Kauffman polynomial -z^2 a^{16}-2 z^3 a^{15}-3 z^4 a^{14}+2 z^2 a^{14}-4 z^5 a^{13}+7 z^3 a^{13}-2 z a^{13}-4 z^6 a^{12}+9 z^4 a^{12}-2 z^2 a^{12}-4 z^7 a^{11}+12 z^5 a^{11}-5 z^3 a^{11}-z a^{11}-4 z^8 a^{10}+17 z^6 a^{10}-19 z^4 a^{10}+7 z^2 a^{10}-a^{10}-3 z^9 a^9+15 z^7 a^9-22 z^5 a^9+13 z^3 a^9-6 z a^9+a^9 z^{-1} -z^{10} a^8+2 z^8 a^8+12 z^6 a^8-34 z^4 a^8+23 z^2 a^8-3 a^8-4 z^9 a^7+27 z^7 a^7-61 z^5 a^7+56 z^3 a^7-22 z a^7+3 a^7 z^{-1} -z^{10} a^6+6 z^8 a^6-9 z^6 a^6-3 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5+8 z^7 a^5-23 z^5 a^5+29 z^3 a^5-15 z a^5+2 a^5 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 \
-2           11
-4            0
-6         31 2
-8        11  0
-10       32   1
-12      21    -1
-14     33     0
-16    23      1
-18   22       0
-20  12        1
-22 12         -1
-24 1          1
-261           -1
Integral Khovanov Homology

(db, data source)

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