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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 L11a424's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,7,17,8 X20,9,21,10 X18,22,19,21 X8,15,9,16 X10,19,5,20 X22,14,11,13 X14,18,15,17 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 4, -7}, {11, -2, 8, -9, 6, -3, 9, -5, 7, -4, 5, -8}
A Braid Representative
A Morse Link Presentation L11a424 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+3 t(1) t(2)^2-3 t(1) t(3) t(2)^2+5 t(3) t(2)^2-3 t(2)^2-3 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-5 t(1) t(2)+8 t(1) t(3) t(2)-8 t(3) t(2)+3 t(2)+3 t(1) t(3)^2-3 t(3)^2+2 t(1)-5 t(1) t(3)+3 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial  q^{-9} -2 q^{-8} +7 q^{-7} -11 q^{-6} +18 q^{-5} -20 q^{-4} +22 q^{-3} -q^2-20 q^{-2} +5 q+15 q^{-1} -10 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^{10} z^{-2} -2 a^8 z^{-2} -4 a^8+6 a^6 z^2+a^6 z^{-2} +5 a^6-4 a^4 z^4-5 a^4 z^2-2 a^4+a^2 z^6+2 a^2 z^4+4 a^2 z^2+a^2-z^4 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+6 a^{10} z^2+a^{10} z^{-2} -4 a^{10}+2 a^9 z^7-4 a^9 z^5+4 a^9 z-2 a^9 z^{-1} +3 a^8 z^8-3 a^8 z^6-5 a^8 z^4+9 a^8 z^2+2 a^8 z^{-2} -6 a^8+3 a^7 z^9-7 a^7 z^5+5 a^7 z^3-2 a^7 z^{-1} +a^6 z^{10}+11 a^6 z^8-26 a^6 z^6+19 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -3 a^6+8 a^5 z^9-a^5 z^7-24 a^5 z^5+24 a^5 z^3-6 a^5 z+a^4 z^{10}+18 a^4 z^8-37 a^4 z^6+21 a^4 z^4-2 a^4 z^2-a^4+5 a^3 z^9+11 a^3 z^7-37 a^3 z^5+24 a^3 z^3-2 a^3 z+10 a^2 z^8-10 a^2 z^6-4 a^2 z^4+3 a^2 z^2-a^2+10 a z^7-15 a z^5+z^5 a^{-1} +5 a z^3+5 z^6-5 z^4 (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 \
5           1-1
3          4 4
1         61 -5
-1        94  5
-3       127   -5
-5      108    2
-7     1012     2
-9    810      -2
-11   411       7
-13  37        -4
-15 16         5
-17 1          -1
-191           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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See/edit the Link Page master template (intermediate).

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