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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 L10a173's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,17,14 X16,19,11,20 X18,7,19,8 X8,16,9,15 X14,10,15,9 X10,17,5,18 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {8, -5, 4, -3}, {9, -1, 5, -6, 7, -8}, {10, -2, 3, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L10a173 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2) t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-2 t(2) t(4) t(3)^2+2 t(4) t(3)^2-t(3)^2-t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+3 t(1) t(4) t(3)-4 t(1) t(2) t(4) t(3)+3 t(2) t(4) t(3)-4 t(4) t(3)+t(3)+t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+t(1)-2 t(1) t(4)+2 t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial \frac{15}{q^{9/2}}-\frac{17}{q^{7/2}}+\frac{11}{q^{5/2}}-\frac{9}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{16}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-3} +a^9 z+a^9 z^{-1} -3 a^7 z^3-3 a^7 z^{-3} -5 a^7 z-6 a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+3 a^5 z^{-3} +9 a^5 z+9 a^5 z^{-1} +a^3 z^5-a^3 z^3-a^3 z^{-3} -5 a^3 z-4 a^3 z^{-1} -a z^3 (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-12 a^9 z^5+14 a^9 z^3-a^9 z^{-3} -12 a^9 z+5 a^9 z^{-1} +5 a^8 z^8-a^8 z^6-13 a^8 z^4+17 a^8 z^2+3 a^8 z^{-2} -10 a^8+2 a^7 z^9+12 a^7 z^7-36 a^7 z^5+42 a^7 z^3-3 a^7 z^{-3} -29 a^7 z+12 a^7 z^{-1} +12 a^6 z^8-17 a^6 z^6-4 a^6 z^4+23 a^6 z^2+6 a^6 z^{-2} -19 a^6+2 a^5 z^9+14 a^5 z^7-40 a^5 z^5+39 a^5 z^3-3 a^5 z^{-3} -23 a^5 z+12 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+7 a^4 z^2+3 a^4 z^{-2} -10 a^4+8 a^3 z^7-16 a^3 z^5+12 a^3 z^3-a^3 z^{-3} -7 a^3 z+5 a^3 z^{-1} +4 a^2 z^6-5 a^2 z^4+a z^5-a z^3 (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 \
2          11
0         3 -3
-2        61 5
-4       64  -2
-6      115   6
-8     810    2
-10    87     1
-12   510      5
-14  36       -3
-16  5        5
-1813         -2
-201          1
Integral Khovanov Homology

(db, data source)

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