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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 L11a457's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,9,13,10 X16,21,17,22 X8,18,9,17 X10,15,11,16 X12,20,5,19 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 5, -8, 3, -9}, {11, -2, 8, -6, 7, -4, 9, -3, 6, -5}
A Braid Representative
A Morse Link Presentation L11a457 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(3)^2 t(2)^3+t(1) t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-3 t(1) t(2)^2+6 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+3 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+2 t(1) t(2)-5 t(1) t(3) t(2)+4 t(3) t(2)-t(2)+t(1) t(3)^3-t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^3-4 q^2+10 q-15+21 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^8+3 a^6 z^2+2 a^6-3 a^4 z^4-2 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^6-3 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -5 a^2-2 z^4+ z^{-2} +3 (db)
Kauffman polynomial a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-5 a^8 z^4+3 a^8 z^2-a^8+5 a^7 z^7-5 a^7 z^5+a^7 z+7 a^6 z^8-9 a^6 z^6+6 a^6 z^4+6 a^5 z^9-3 a^5 z^7-6 a^5 z^5+10 a^5 z^3-3 a^5 z+2 a^4 z^{10}+15 a^4 z^8-45 a^4 z^6+52 a^4 z^4-30 a^4 z^2-a^4 z^{-2} +9 a^4+13 a^3 z^9-21 a^3 z^7+3 a^3 z^5+10 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+16 a^2 z^8-53 a^2 z^6+z^6 a^{-2} +59 a^2 z^4-2 z^4 a^{-2} -39 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+7 a z^9-9 a z^7+4 z^7 a^{-1} -5 a z^5-8 z^5 a^{-1} +6 a z^3+4 z^3 a^{-1} -5 a z+2 a z^{-1} +8 z^8-19 z^6+16 z^4-11 z^2- z^{-2} +6 (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          3 -3
3         71 6
1        83  -5
-1       137   6
-3      1211    -1
-5     1110     1
-7    812      4
-9   611       -5
-11  29        7
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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