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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 L11a462's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X16,5,17,6 X18,14,19,13 X12,15,5,16 X10,19,11,20 X8,21,9,22 X20,7,21,8 X22,9,13,10 X2,11,3,12 X4,18,1,17
Gauss code {1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}
A Braid Representative
A Morse Link Presentation L11a462 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+6 t(1) t(3)^2 t(2)^2-6 t(3)^2 t(2)^2+t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-t(3)^3 t(2)-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-2 t(1) t(2)+6 t(1) t(3) t(2)-6 t(3) t(2)+2 t(2)-t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +21 q^{-4} -21 q^{-5} +19 q^{-6} -14 q^{-7} +10 q^{-8} -4 q^{-9} + q^{-10} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^4+2 a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^6-7 a^6 z^4-8 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^8+5 a^4 z^6+9 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-3 a^2 z^4-a^2 z^2+2 a^2 (db)
Kauffman polynomial a^{12} z^4+4 a^{11} z^5+10 a^{10} z^6-10 a^{10} z^4+6 a^{10} z^2-2 a^{10}+14 a^9 z^7-18 a^9 z^5+6 a^9 z^3+a^9 z+13 a^8 z^8-17 a^8 z^6+6 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +3 a^8+7 a^7 z^9+4 a^7 z^7-32 a^7 z^5+22 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+16 a^6 z^8-50 a^6 z^6+42 a^6 z^4-22 a^6 z^2-2 a^6 z^{-2} +9 a^6+12 a^5 z^9-26 a^5 z^7+4 a^5 z^5+14 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+7 a^4 z^8-38 a^4 z^6+42 a^4 z^4-14 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-15 a^3 z^7+11 a^3 z^5+a^3 z+4 a^2 z^8-15 a^2 z^6+17 a^2 z^4-4 a^2 z^2-2 a^2+a z^7-3 a z^5+2 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 \
3           1-1
1          3 3
-1         41 -3
-3        93  6
-5       106   -4
-7      117    4
-9     1010     0
-11    911      -2
-13   611       5
-15  48        -4
-17  6         6
-1914          -3
-211           1
Integral Khovanov Homology

(db, data source)

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