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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 L11a390's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X18,12,19,11 X20,13,21,14 X22,20,9,19 X12,21,13,22 X14,18,15,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L11a390 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^3-4 u v w^2+6 u v w-2 u v-2 u w^3+7 u w^2-7 u w+2 u-2 v w^3+7 v w^2-7 v w+2 v+2 w^3-6 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial  q^{-9} -2 q^{-8} +6 q^{-7} -10 q^{-6} +16 q^{-5} -19 q^{-4} +21 q^{-3} -q^2-18 q^{-2} +5 q+15 q^{-1} -10 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^{10} z^{-2} -3 a^8 z^{-2} -4 a^8+6 a^6 z^2+4 a^6 z^{-2} +8 a^6-4 a^4 z^4-7 a^4 z^2-3 a^4 z^{-2} -7 a^4+a^2 z^6+2 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +3 a^2-z^4 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+6 a^{10} z^2+a^{10} z^{-2} -4 a^{10}+2 a^9 z^7-5 a^9 z^5+3 a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^8+2 a^8 z^6-18 a^8 z^4+25 a^8 z^2+3 a^8 z^{-2} -14 a^8+2 a^7 z^9+3 a^7 z^7-11 a^7 z^5+6 a^7 z^3+a^7 z-a^7 z^{-1} +a^6 z^{10}+6 a^6 z^8-5 a^6 z^6-19 a^6 z^4+35 a^6 z^2+4 a^6 z^{-2} -21 a^6+7 a^5 z^9-3 a^5 z^7-12 a^5 z^5+10 a^5 z^3+a^5 z-a^5 z^{-1} +a^4 z^{10}+14 a^4 z^8-23 a^4 z^6-3 a^4 z^4+23 a^4 z^2+3 a^4 z^{-2} -14 a^4+5 a^3 z^9+6 a^3 z^7-22 a^3 z^5+10 a^3 z^3+a^3 z-a^3 z^{-1} +10 a^2 z^8-12 a^2 z^6-3 a^2 z^4+7 a^2 z^2+a^2 z^{-2} -4 a^2+10 a z^7-15 a z^5+z^5 a^{-1} +3 a z^3+5 z^6-5 z^4 (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          4 4
1         61 -5
-1        94  5
-3       107   -3
-5      118    3
-7     810     2
-9    811      -3
-11   511       6
-13  15        -4
-15 15         4
-17 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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See/edit the Link Page master template (intermediate).

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