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

Visit L10a74 at Knotilus!

Link Presentations

[edit Notes on L10a74's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,11,19,12 X20,13,7,14 X12,19,13,20 X14,17,15,18 X6718 X4,15,5,16
Gauss code {1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -7, 6, -8, 10, -4, 8, -5, 7, -6}
A Braid Representative
A Morse Link Presentation L10a74 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+2}{u v^2} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{3}{q^{25/2}}-\frac{5}{q^{23/2}}+\frac{7}{q^{21/2}}-\frac{8}{q^{19/2}}+\frac{8}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{11} z^5+3 a^{11} z^3+a^{11} z-a^{11} z^{-1} -a^9 z^7-4 a^9 z^5-2 a^9 z^3+5 a^9 z+3 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-12 a^7 z^3-9 a^7 z-2 a^7 z^{-1} (db)
Kauffman polynomial a^{17} z^3+3 a^{16} z^4-a^{16} z^2+5 a^{15} z^5-4 a^{15} z^3+a^{15} z+6 a^{14} z^6-8 a^{14} z^4+3 a^{14} z^2+5 a^{13} z^7-7 a^{13} z^5+a^{13} z+3 a^{12} z^8-3 a^{12} z^6-5 a^{12} z^4+3 a^{12} z^2-a^{12}+a^{11} z^9+2 a^{11} z^7-10 a^{11} z^5+4 a^{11} z^3+a^{11} z^{-1} +4 a^{10} z^8-12 a^{10} z^6+5 a^{10} z^4+6 a^{10} z^2-3 a^{10}+a^9 z^9-2 a^9 z^7-4 a^9 z^5+11 a^9 z^3-9 a^9 z+3 a^9 z^{-1} +a^8 z^8-3 a^8 z^6-a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^7-6 a^7 z^5+12 a^7 z^3-9 a^7 z+2 a^7 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 \
-6          11
-8         110
-10        3  3
-12       21  -1
-14      53   2
-16     32    -1
-18    55     0
-20   34      1
-22  24       -2
-24 13        2
-26 2         -2
-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}
r=-9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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