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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a117 at Knotilus!

Link Presentations

[edit Notes on L10a117's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X20,9,11,10 X16,5,17,6 X18,7,19,8 X6,17,7,18 X8,19,9,20 X10,15,1,16
Gauss code {1, -2, 3, -4, 6, -8, 7, -9, 5, -10}, {4, -1, 2, -3, 10, -6, 8, -7, 9, -5}
A Braid Representative
A Morse Link Presentation L10a117 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^4 \left(-v^3\right)-u^3 v^4+2 u^3 v^3-2 u^3 v^2-2 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^2+2 u v-u-v}{u^2 v^2} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-7 z^3 a^9-3 z a^9-a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-6 z a^7 (db)
Kauffman polynomial -z^3 a^{17}+z a^{17}-2 z^4 a^{16}+z^2 a^{16}-3 z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+6 z^5 a^{13}-5 z^3 a^{13}+2 z a^{13}-2 z^8 a^{12}+4 z^6 a^{12}-2 z^4 a^{12}+2 z^2 a^{12}-z^9 a^{11}+2 z^7 a^{11}-2 z^5 a^{11}+7 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} -3 z^8 a^{10}+11 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}-a^{10}-z^9 a^9+4 z^7 a^9-5 z^5 a^9+5 z^3 a^9-4 z a^9+a^9 z^{-1} -z^8 a^8+4 z^6 a^8-3 z^4 a^8-z^2 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+6 z a^7 (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 \
-6          11
-8         110
-10        2  2
-12       21  -1
-14      32   1
-16     32    -1
-18    33     0
-20   23      1
-22  23       -1
-24  2        2
-2612         -1
-281          1
Integral Khovanov Homology

(db, data source)

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