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

Visit L11a351 at Knotilus!

Link Presentations

[edit Notes on L11a351's Link Presentations]

Planar diagram presentation X12,1,13,2 X18,9,19,10 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X8394 X16,21,17,22 X4,18,5,17 X10,13,1,14 X2,19,3,20
Gauss code {1, -11, 7, -9, 3, -4, 6, -7, 2, -10}, {4, -1, 10, -3, 5, -8, 9, -2, 11, -6, 8, -5}
A Braid Representative
A Morse Link Presentation L11a351 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+2 t(1)^2-2 t(2)^2 t(1)+7 t(2) t(1)-2 t(1)+2 t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{7/2}+5 q^{5/2}-13 q^{3/2}+20 \sqrt{q}-\frac{27}{\sqrt{q}}+\frac{30}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{25}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+3 a^5 z^3+2 a^5 z-3 a^3 z^5-5 a^3 z^3-3 a^3 z+a z^7+3 a z^5-z^5 a^{-1} +6 a z^3-z^3 a^{-1} +4 a z+a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+5 a^7 z^3-a^7 z+8 a^6 z^8-16 a^6 z^6+12 a^6 z^4-4 a^6 z^2+9 a^5 z^9-13 a^5 z^7+3 a^5 z^5+2 a^5 z^3+4 a^4 z^{10}+14 a^4 z^8-44 a^4 z^6+33 a^4 z^4-7 a^4 z^2+22 a^3 z^9-36 a^3 z^7+9 a^3 z^5+z^5 a^{-3} +3 a^3 z^3+4 a^2 z^{10}+24 a^2 z^8-59 a^2 z^6+5 z^6 a^{-2} +33 a^2 z^4-2 z^4 a^{-2} -4 a^2 z^2+13 a z^9-6 a z^7+13 z^7 a^{-1} -19 a z^5-16 z^5 a^{-1} +13 a z^3+7 z^3 a^{-1} -4 a z-3 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +18 z^8-27 z^6+12 z^4-2 z^2-1 (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          4 -4
4         91 8
2        114  -7
0       169   7
-2      1613    -3
-4     1414     0
-6    1116      5
-8   714       -7
-10  311        8
-12 17         -6
-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}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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