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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 L11a230's Link Presentations]

Planar diagram presentation X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X6718 X4,15,5,16 X22,20,7,19 X20,10,21,9 X10,22,11,21
Gauss code {1, -2, 3, -8, 4, -7}, {7, -1, 10, -11, 2, -3, 5, -6, 8, -4, 6, -5, 9, -10, 11, -9}
A Braid Representative
A Morse Link Presentation L11a230 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-5 u^2 v^3+6 u^2 v^2-2 u^2 v-u v^4+7 u v^3-11 u v^2+7 u v-u-2 v^3+6 v^2-5 v+2}{u v^2} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-4 z^5 a^5-6 z^3 a^5-2 z a^5+2 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-7 z^3 a^3-7 z a^3-3 a^3 z^{-1} +z^5 a+3 z^3 a+3 z a+a z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+7 z^5 a^9-3 z^3 a^9+z a^9-6 z^8 a^8+9 z^6 a^8-8 z^4 a^8+4 z^2 a^8-4 z^9 a^7+6 z^5 a^7-4 z^3 a^7-z a^7-z^{10} a^6-11 z^8 a^6+26 z^6 a^6-24 z^4 a^6+9 z^2 a^6-7 z^9 a^5+6 z^7 a^5+8 z^5 a^5-10 z^3 a^5+4 z a^5-2 a^5 z^{-1} -z^{10} a^4-9 z^8 a^4+21 z^6 a^4-10 z^4 a^4-2 z^2 a^4+3 a^4-3 z^9 a^3-2 z^7 a^3+18 z^5 a^3-18 z^3 a^3+10 z a^3-3 a^3 z^{-1} -4 z^8 a^2+6 z^6 a^2+4 z^4 a^2-8 z^2 a^2+3 a^2-3 z^7 a+8 z^5 a-7 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+1 (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          2 2
0         51 -4
-2        72  5
-4       96   -3
-6      106    4
-8     89     1
-10    810      -2
-12   48       4
-14  38        -5
-16 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\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}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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