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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 L11n346's Link Presentations]

Planar diagram presentation X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4
Gauss code {1, -10, -2, 11}, {10, -1, -4, 9, -6, 7, -3, 8}, {-11, 2, -7, 6, -5, 4, -8, 3, -9, 5}
A Braid Representative
A Morse Link Presentation L11n346 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^3-t(1) t(3) t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-3 t(1) t(3)^2 t(2)^2+3 t(3)^2 t(2)^2-2 t(1) t(2)^2+4 t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+2 t(3)^3 t(2)+2 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-3 t(1) t(3) t(2)+3 t(3) t(2)-t(2)-t(3)^3+t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^3-3 q^2+7 q-10+13 q^{-1} -12 q^{-2} +13 q^{-3} -9 q^{-4} +6 q^{-5} -2 q^{-6} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^6-z^4 a^4+z^2 a^4+a^4 z^{-2} +3 a^4+z^6 a^2+2 z^4 a^2-2 a^2 z^{-2} -3 a^2-2 z^4-3 z^2+ z^{-2} +z^2 a^{-2} + a^{-2} (db)
Kauffman polynomial 2 a^3 z^9+2 a z^9+6 a^4 z^8+10 a^2 z^8+4 z^8+5 a^5 z^7+6 a^3 z^7+4 a z^7+3 z^7 a^{-1} +a^6 z^6-15 a^4 z^6-25 a^2 z^6+z^6 a^{-2} -8 z^6-9 a^5 z^5-22 a^3 z^5-21 a z^5-8 z^5 a^{-1} +6 a^6 z^4+27 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +z^4+3 a^7 z^3+15 a^5 z^3+23 a^3 z^3+17 a z^3+6 z^3 a^{-1} -6 a^6 z^2-26 a^4 z^2-23 a^2 z^2+3 z^2 a^{-2} -2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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 \
7         11
5        2 -2
3       51 4
1      52  -3
-1     85   3
-3    78    1
-5   65     1
-7  37      4
-9 36       -3
-11 4        4
-132         -2
Integral Khovanov Homology

(db, data source)

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