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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 L11a268's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X16,14,17,13 X14,5,15,6 X4,15,5,16 X6,22,7,21 X18,7,19,8 X20,17,21,18 X8,19,1,20
Gauss code {1, -4, 2, -7, 6, -8, 9, -11}, {4, -1, 3, -2, 5, -6, 7, -5, 10, -9, 11, -10, 8, -3}
A Braid Representative
A Morse Link Presentation L11a268 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-u^3-3 u^2 v^3+9 u^2 v^2-10 u^2 v+3 u^2+3 u v^3-10 u v^2+9 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+4 q^{5/2}-\frac{21}{q^{5/2}}-8 q^{3/2}+\frac{21}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+13 \sqrt{q}-\frac{19}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+3 z^3 a^5+4 z a^5+a^5 z^{-1} -3 z^5 a^3-8 z^3 a^3-7 z a^3-a^3 z^{-1} +z^7 a+4 z^5 a+7 z^3 a+3 z a-z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-8 a^3 z^9-4 a z^9-6 a^6 z^8-15 a^4 z^8-16 a^2 z^8-7 z^8-4 a^7 z^7-4 a^5 z^7-3 a^3 z^7-10 a z^7-7 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+36 a^4 z^6+32 a^2 z^6-4 z^6 a^{-2} +5 z^6+10 a^7 z^5+29 a^5 z^5+40 a^3 z^5+33 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-5 a^6 z^4-22 a^4 z^4-17 a^2 z^4+6 z^4 a^{-2} +4 z^4-8 a^7 z^3-29 a^5 z^3-42 a^3 z^3-29 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+3 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+2 a^7 z+11 a^5 z+14 a^3 z+7 a z+2 z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-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 \
8           11
6          3 -3
4         51 4
2        83  -5
0       115   6
-2      119    -2
-4     1010     0
-6    811      3
-8   610       -4
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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