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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 L11a526's Link Presentations]

Planar diagram presentation X8192 X14,3,15,4 X12,15,7,16 X10,22,11,21 X16,9,17,10 X20,12,21,11 X2738 X22,17,13,18 X6,13,1,14 X4,20,5,19 X18,6,19,5
Gauss code {1, -7, 2, -10, 11, -9}, {7, -1, 5, -4, 6, -3}, {9, -2, 3, -5, 8, -11, 10, -6, 4, -8}
A Braid Representative
A Morse Link Presentation L11a526 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(-t(2)^2 t(1)^2+t(2) t(3)^2 t(1)^2-t(3)^2 t(1)^2+2 t(2) t(1)^2-2 t(2) t(3) t(1)^2+t(3) t(1)^2-t(1)^2+2 t(2)^2 t(1)+t(2)^2 t(3)^2 t(1)-3 t(2) t(3)^2 t(1)+2 t(3)^2 t(1)-3 t(2) t(1)-2 t(2)^2 t(3) t(1)+6 t(2) t(3) t(1)-2 t(3) t(1)+t(1)-t(2)^2-t(2)^2 t(3)^2+2 t(2) t(3)^2-t(3)^2+t(2)+t(2)^2 t(3)-2 t(2) t(3)\right)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^3-4 q^2+10 q-16+23 q^{-1} -25 q^{-2} +27 q^{-3} -22 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 \left(-z^4\right)-a^6 z^2-2 a^6+a^4 z^6+2 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +6 a^4+a^2 z^6-a^2 z^4-6 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -7 a^2-2 z^4+ z^{-2} +3 (db)
Kauffman polynomial a^9 z^5-a^9 z^3+4 a^8 z^6-4 a^8 z^4+9 a^7 z^7-13 a^7 z^5+7 a^7 z^3-2 a^7 z+13 a^6 z^8-25 a^6 z^6+24 a^6 z^4-13 a^6 z^2+4 a^6+10 a^5 z^9-10 a^5 z^7-5 a^5 z^5+13 a^5 z^3-6 a^5 z+3 a^4 z^{10}+20 a^4 z^8-64 a^4 z^6+73 a^4 z^4-42 a^4 z^2-a^4 z^{-2} +13 a^4+18 a^3 z^9-35 a^3 z^7+18 a^3 z^5+3 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^{10}+15 a^2 z^8-54 a^2 z^6+z^6 a^{-2} +62 a^2 z^4-2 z^4 a^{-2} -40 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+8 a z^9-12 a z^7+4 z^7 a^{-1} +a z^5-8 z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -3 a z+2 a z^{-1} +8 z^8-18 z^6+15 z^4-10 z^2- z^{-2} +5 (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 \
7           11
5          3 -3
3         71 6
1        93  -6
-1       147   7
-3      1311    -2
-5     1412     2
-7    1015      5
-9   712       -5
-11  310        7
-13 17         -6
-15 3          3
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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