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

Visit L10a97 at Knotilus!

Link Presentations

[edit Notes on L10a97's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,7,19,8 X20,16,9,15 X16,20,17,19 X8,17,1,18 X4,13,5,14
Gauss code {1, -2, 3, -10, 4, -5, 6, -9}, {5, -1, 2, -3, 10, -4, 7, -8, 9, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L10a97 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{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+3 u v-2 u-2 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial z^3 a^9+3 z a^9+a^9 z^{-1} -2 z^5 a^7-8 z^3 a^7-7 z a^7-a^7 z^{-1} +z^7 a^5+5 z^5 a^5+7 z^3 a^5+3 z a^5-z^5 a^3-4 z^3 a^3-3 z a^3 (db)
Kauffman polynomial -z^3 a^{13}+z a^{13}-2 z^4 a^{12}+z^2 a^{12}-3 z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-3 z^6 a^{10}+3 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+6 z^5 a^9-6 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+3 z^6 a^8+2 z^4 a^8-2 z^2 a^8+a^8-z^9 a^7+9 z^5 a^7-12 z^3 a^7+7 z a^7-a^7 z^{-1} -4 z^8 a^6+16 z^6 a^6-17 z^4 a^6+5 z^2 a^6-z^9 a^5+2 z^7 a^5+5 z^5 a^5-9 z^3 a^5+2 z a^5-2 z^8 a^4+10 z^6 a^4-14 z^4 a^4+5 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3 (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 \
0          11
-2         1 -1
-4        21 1
-6       32  -1
-8      31   2
-10     33    0
-12    33     0
-14   23      1
-16  23       -1
-18  2        2
-2012         -1
-221          1
Integral Khovanov Homology

(db, data source)

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