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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 L11a522's Link Presentations]

Planar diagram presentation X8192 X14,4,15,3 X16,5,17,6 X18,8,19,7 X12,20,7,19 X10,14,11,13 X22,10,13,9 X20,15,21,16 X6,17,1,18 X2,11,3,12 X4,22,5,21
Gauss code {1, -10, 2, -11, 3, -9}, {4, -1, 7, -6, 10, -5}, {6, -2, 8, -3, 9, -4, 5, -8, 11, -7}
A Braid Representative
A Morse Link Presentation L11a522 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2 w^3-2 u^2 v^2 w^2+u^2 v^2 w-2 u^2 v w^3+4 u^2 v w^2-3 u^2 v w+u^2 v+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-4 u v^2 w+u v^2+3 u v w^3-8 u v w^2+8 u v w-3 u v-u w^3+4 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2-v w^3+3 v w^2-4 v w+2 v-w^2+2 w-1}{u v w^{3/2}} (db)
Jones polynomial q^6-4 q^5+10 q^4-15 q^3+22 q^2-24 q+25-21 q^{-1} +16 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-9 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^8+5 z^6+10 z^4+8 z^2+ z^{-2} +3 (db)
Kauffman polynomial 3 z^{10} a^{-2} +3 z^{10}+9 a z^9+17 z^9 a^{-1} +8 z^9 a^{-3} +11 a^2 z^8+13 z^8 a^{-2} +8 z^8 a^{-4} +16 z^8+8 a^3 z^7-10 a z^7-36 z^7 a^{-1} -14 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-52 z^6 a^{-2} -19 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-11 a^3 z^5-2 a z^5+21 z^5 a^{-1} +3 z^5 a^{-3} -8 z^5 a^{-5} -5 a^4 z^4+15 a^2 z^4+62 z^4 a^{-2} +16 z^4 a^{-4} -2 z^4 a^{-6} +64 z^4-a^5 z^3+4 a^3 z^3+9 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +3 z^3 a^{-5} +a^4 z^2-7 a^2 z^2-37 z^2 a^{-2} -12 z^2 a^{-4} +z^2 a^{-6} -32 z^2-a^3 z-3 a z-8 z a^{-1} -6 z a^{-3} +a^2+12 a^{-2} +6 a^{-4} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} 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 \
13           11
11          3 -3
9         71 6
7        94  -5
5       136   7
3      1210    -2
1     1312     1
-1    913      4
-3   712       -5
-5  310        7
-7 16         -5
-9 3          3
-111           -1
Integral Khovanov Homology

(db, data source)

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