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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 L11a449's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X22,18,13,17 X16,8,17,7 X12,14,5,13 X8,21,9,22 X20,11,21,12 X18,9,19,10 X10,19,11,20 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -6, 8, -9, 7, -5}, {5, -2, 11, -4, 3, -8, 9, -7, 6, -3}
A Braid Representative
A Morse Link Presentation L11a449 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-4 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+5 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-5 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^5+3 q^4-6 q^3+10 q^2-13 q+17-15 q^{-1} +14 q^{-2} -10 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-14 a^2 z^2-5 z^2 a^{-2} +15 z^2+3 a^4-9 a^2-2 a^{-2} +8+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-2 z^3 a^{-5} -a^5 z+4 a^4 z^8-8 a^4 z^6+3 z^6 a^{-4} +a^4 z^4-6 z^4 a^{-4} -2 a^4 z^2+z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 a^3 z^9-a^3 z^7+5 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +14 a^3 z^3+7 z^3 a^{-3} -8 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -30 a^2 z^6-16 z^6 a^{-2} +39 a^2 z^4+21 z^4 a^{-2} -28 a^2 z^2-13 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2+3 a^{-2} +7 a z^9+4 z^9 a^{-1} -14 a z^7-5 z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +16 a z^3+16 z^3 a^{-1} -12 a z-7 z a^{-1} +2 a z^{-1} +z^{10}+10 z^8-40 z^6+62 z^4-37 z^2- z^{-2} +11 (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 \
11           1-1
9          2 2
7         41 -3
5        62  4
3       74   -3
1      106    4
-1     810     2
-3    67      -1
-5   48       4
-7  36        -3
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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