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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 L11a527's Link Presentations]

Planar diagram presentation X8192 X14,3,15,4 X12,15,7,16 X10,21,11,22 X16,9,17,10 X22,11,13,12 X18,6,19,5 X20,18,21,17 X2738 X4,13,5,14 X6,20,1,19
Gauss code {1, -9, 2, -10, 7, -11}, {9, -1, 5, -4, 6, -3}, {10, -2, 3, -5, 8, -7, 11, -8, 4, -6}
A Braid Representative
A Morse Link Presentation L11a527 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2 w^2-u^2 v^2 w+u^2 v w^3-4 u^2 v w^2+4 u^2 v w-u^2 v+2 u^2 w^2-3 u^2 w+u^2+u v^2 w^3-3 u v^2 w^2+3 u v^2 w-2 u v w^3+7 u v w^2-7 u v w+2 u v-3 u w^2+3 u w-u-v^2 w^3+3 v^2 w^2-2 v^2 w+v w^3-4 v w^2+4 v w-v+w^2-w}{u v w^{3/2}} (db)
Jones polynomial  q^{-9} -3 q^{-8} +8 q^{-7} -13 q^{-6} +19 q^{-5} -21 q^{-4} +23 q^{-3} -q^2-19 q^{-2} +4 q+15 q^{-1} -9 (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8 z^{-2} +2 a^8-3 z^4 a^6-8 z^2 a^6-2 a^6 z^{-2} -8 a^6+2 z^6 a^4+7 z^4 a^4+11 z^2 a^4+a^4 z^{-2} +7 a^4+z^6 a^2+z^4 a^2-z^2 a^2-a^2-z^4-z^2 (db)
Kauffman polynomial a^{10} z^6-3 a^{10} z^4+3 a^{10} z^2-a^{10}+3 a^9 z^7-7 a^9 z^5+4 a^9 z^3+5 a^8 z^8-10 a^8 z^6+6 a^8 z^4-5 a^8 z^2-a^8 z^{-2} +4 a^8+5 a^7 z^9-6 a^7 z^7-3 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+10 a^6 z^8-35 a^6 z^6+43 a^6 z^4-31 a^6 z^2-2 a^6 z^{-2} +12 a^6+12 a^5 z^9-21 a^5 z^7+8 a^5 z^5+7 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-45 a^4 z^6+52 a^4 z^4-31 a^4 z^2-a^4 z^{-2} +10 a^4+7 a^3 z^9-4 a^3 z^7-9 a^3 z^5+11 a^3 z^3-3 a^3 z+10 a^2 z^8-17 a^2 z^6+13 a^2 z^4-7 a^2 z^2+2 a^2+8 a z^7-12 a z^5+z^5 a^{-1} +5 a z^3-z^3 a^{-1} -a z+4 z^6-5 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 \
5           1-1
3          3 3
1         61 -5
-1        93  6
-3       117   -4
-5      128    4
-7     1012     2
-9    911      -2
-11   511       6
-13  38        -5
-15 16         5
-17 2          -2
-191           1
Integral Khovanov Homology

(db, data source)

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