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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 L11a257's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X14,5,15,6 X20,16,21,15 X16,8,17,7 X6,21,7,22 X18,9,19,10 X22,17,9,18 X8,20,1,19 X4,11,5,12 X2,13,3,14
Gauss code {1, -11, 2, -10, 3, -6, 5, -9}, {7, -1, 10, -2, 11, -3, 4, -5, 8, -7, 9, -4, 6, -8}
A Braid Representative
A Morse Link Presentation L11a257 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+4 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+7 t(1)^2 t(2)^3-6 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-6 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+4 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \frac{18}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{16}{q^{5/2}}+q^{3/2}-\frac{13}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{16}{q^{11/2}}-4 \sqrt{q}+\frac{8}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+3 a^7 z^3+2 a^7 z-a^5 z^7-4 a^5 z^5-6 a^5 z^3-3 a^5 z+a^5 z^{-1} -a^3 z^7-3 a^3 z^5-2 a^3 z^3-a^3 z-a^3 z^{-1} +a z^5+2 a z^3 (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-7 z^3 a^9+2 z a^9-6 z^8 a^8+12 z^6 a^8-11 z^4 a^8+2 z^2 a^8-5 z^9 a^7+8 z^7 a^7-6 z^5 a^7-3 z^3 a^7+z a^7-2 z^{10} a^6-5 z^8 a^6+18 z^6 a^6-21 z^4 a^6+6 z^2 a^6-10 z^9 a^5+24 z^7 a^5-25 z^5 a^5+12 z^3 a^5-a^5 z^{-1} -2 z^{10} a^4-5 z^8 a^4+17 z^6 a^4-10 z^4 a^4+2 z^2 a^4+a^4-5 z^9 a^3+7 z^7 a^3+2 z^5 a^3+z a^3-a^3 z^{-1} -6 z^8 a^2+13 z^6 a^2-4 z^4 a^2-z^2 a^2-4 z^7 a+10 z^5 a-6 z^3 a-z^6+2 z^4-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 \
4           1-1
2          3 3
0         51 -4
-2        83  5
-4       96   -3
-6      107    3
-8     89     1
-10    810      -2
-12   49       5
-14  27        -5
-16 14         3
-18 2          -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}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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