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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 L11a307's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,5,17,6 X22,14,9,13 X14,18,15,17 X20,7,21,8 X18,22,19,21 X6,15,7,16 X8,19,1,20
Gauss code {1, -2, 3, -4, 5, -10, 8, -11}, {4, -1, 2, -3, 6, -7, 10, -5, 7, -9, 11, -8, 9, -6}
A Braid Representative
A Morse Link Presentation L11a307 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-3 u^3 v^2+3 u^3 v-3 u^2 v^3+9 u^2 v^2-8 u^2 v+3 u^2+3 u v^3-8 u v^2+9 u v-3 u+3 v^2-3 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{3/2}-4 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+5 z a^5-z^7 a^3-4 z^5 a^3-7 z^3 a^3-5 z a^3+z^5 a+2 z^3 a (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-11 a^9 z^5+11 a^9 z^3-7 a^9 z+a^9 z^{-1} +6 a^8 z^8-5 a^8 z^6-3 a^8 z^4+4 a^8 z^2-a^8+4 a^7 z^9+3 a^7 z^7-14 a^7 z^5+12 a^7 z^3-5 a^7 z+a^7 z^{-1} +a^6 z^{10}+13 a^6 z^8-31 a^6 z^6+27 a^6 z^4-9 a^6 z^2+8 a^5 z^9-8 a^5 z^7-8 a^5 z^5+10 a^5 z^3-a^5 z+a^4 z^{10}+13 a^4 z^8-40 a^4 z^6+41 a^4 z^4-16 a^4 z^2+4 a^3 z^9-a^3 z^7-17 a^3 z^5+18 a^3 z^3-4 a^3 z+6 a^2 z^8-16 a^2 z^6+13 a^2 z^4-4 a^2 z^2+4 a z^7-11 a z^5+7 a z^3+z^6-2 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 \
4           1-1
2          3 3
0         41 -3
-2        83  5
-4       105   -5
-6      97    2
-8     1010     0
-10    79      -2
-12   510       5
-14  37        -4
-16  5         5
-1813          -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} {\mathbb Z}
r=-7 {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
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

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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