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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 L11a491's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,11,17,12 X20,8,21,7 X8,20,9,19 X14,21,15,22 X22,15,19,16 X18,13,5,14 X12,17,13,18 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {5, -4, 6, -7}, {10, -1, 4, -5, 11, -2, 3, -9, 8, -6, 7, -3, 9, -8}
A Braid Representative
A Morse Link Presentation L11a491 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-1) (t(3)-1) \left(t(3)^4+2 t(1) t(3)^3-2 t(3)^3-2 t(1) t(3)^2+2 t(3)^2+2 t(1) t(3)-2 t(3)-t(1)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} (db)
Jones polynomial - q^{-11} +3 q^{-10} -8 q^{-9} +13 q^{-8} -16 q^{-7} +19 q^{-6} -17 q^{-5} +16 q^{-4} -10 q^{-3} +6 q^{-2} -2 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+3 a^8 z^4+8 a^8 z^2+4 a^8 z^{-2} +8 a^8-2 a^6 z^6-7 a^6 z^4-10 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-2 a^4 z^4+2 a^4 z^{-2} +2 a^4+a^2 z^4+3 a^2 z^2+2 a^2 (db)
Kauffman polynomial z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +7 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-11 z^2 a^{10}-a^{10} z^{-2} +6 a^{10}+4 z^9 a^9+5 z^7 a^9-29 z^5 a^9+41 z^3 a^9-25 z a^9+5 a^9 z^{-1} +z^{10} a^8+13 z^8 a^8-39 z^6 a^8+55 z^4 a^8-45 z^2 a^8-4 a^8 z^{-2} +21 a^8+7 z^9 a^7-6 z^7 a^7-18 z^5 a^7+43 z^3 a^7-35 z a^7+9 a^7 z^{-1} +z^{10} a^6+9 z^8 a^6-32 z^6 a^6+49 z^4 a^6-44 z^2 a^6-5 a^6 z^{-2} +22 a^6+3 z^9 a^5-3 z^7 a^5-5 z^5 a^5+14 z^3 a^5-15 z a^5+5 a^5 z^{-1} +3 z^8 a^4-6 z^6 a^4+4 z^4 a^4-6 z^2 a^4-2 a^4 z^{-2} +6 a^4+2 z^7 a^3-5 z^5 a^3+2 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^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 \
1           11
-1          1 -1
-3         51 4
-5        73  -4
-7       93   6
-9      87    -1
-11     119     2
-13    710      3
-15   69       -3
-17  27        5
-19 16         -5
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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