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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 L11n365's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X9,18,10,19 X17,8,18,9 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -9}
A Braid Representative
A Morse Link Presentation L11n365 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(-v^2 w+v w^3-v w^2+v w-v+w^2\right)}{\sqrt{u} v w^2} (db)
Jones polynomial -1+4 q^{-1} -5 q^{-2} +8 q^{-3} -8 q^{-4} +8 q^{-5} -6 q^{-6} +5 q^{-7} -2 q^{-8} + q^{-9} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^4-6 a^6 z^2-2 a^6 z^{-2} -6 a^6+a^4 z^6+4 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^4-a^2 z^2+a^2 (db)
Kauffman polynomial z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-6 z^5 a^9+3 z^3 a^9+z a^9+2 z^8 a^8-4 z^6 a^8-z^4 a^8-z^2 a^8-a^8 z^{-2} +3 a^8+z^9 a^7+z^7 a^7-9 z^5 a^7+9 z^3 a^7-7 z a^7+2 a^7 z^{-1} +5 z^8 a^6-17 z^6 a^6+25 z^4 a^6-23 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5+z^7 a^5-8 z^5 a^5+14 z^3 a^5-9 z a^5+2 a^5 z^{-1} +3 z^8 a^4-12 z^6 a^4+26 z^4 a^4-21 z^2 a^4-a^4 z^{-2} +7 a^4+2 z^7 a^3-5 z^5 a^3+9 z^3 a^3-2 z a^3+4 z^4 a^2-4 z^2 a^2+z^3 a-z a (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 \
1         1-1
-1        3 3
-3       43 -1
-5      41  3
-7     44   0
-9    44    0
-11   24     2
-13  34      -1
-15 14       3
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\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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