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

Visit L11a317 at Knotilus!

Link Presentations

[edit Notes on L11a317's Link Presentations]

Planar diagram presentation X10,1,11,2 X20,11,21,12 X6,9,7,10 X16,7,17,8 X8,15,1,16 X22,17,9,18 X12,4,13,3 X18,6,19,5 X4,14,5,13 X14,21,15,22 X2,20,3,19
Gauss code {1, -11, 7, -9, 8, -3, 4, -5}, {3, -1, 2, -7, 9, -10, 5, -4, 6, -8, 11, -2, 10, -6}
A Braid Representative
A Morse Link Presentation L11a317 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^3-2 u^2 v^2+u^2 v+u v^4-3 u v^3+6 u v^2-3 u v+u+v^3-2 v^2+v\right)}{u^{3/2} v^{5/2}} (db)
Jones polynomial -\frac{17}{q^{9/2}}-q^{7/2}+\frac{24}{q^{7/2}}+5 q^{5/2}-\frac{29}{q^{5/2}}-12 q^{3/2}+\frac{28}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{10}{q^{11/2}}+19 \sqrt{q}-\frac{26}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+2 a z^5-z^5 a^{-1} -2 a^5 z^3+5 a^3 z^3-z^3 a^{-1} -2 a^5 z+5 a^3 z-3 a z-a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} (db)
Kauffman polynomial -4 a^4 z^{10}-4 a^2 z^{10}-9 a^5 z^9-21 a^3 z^9-12 a z^9-8 a^6 z^8-11 a^4 z^8-19 a^2 z^8-16 z^8-4 a^7 z^7+16 a^5 z^7+40 a^3 z^7+8 a z^7-12 z^7 a^{-1} -a^8 z^6+17 a^6 z^6+38 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5-11 a^5 z^5-26 a^3 z^5+9 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-13 a^6 z^4-32 a^4 z^4-29 a^2 z^4+3 z^4 a^{-2} -9 z^4-4 a^7 z^3+7 a^5 z^3+15 a^3 z^3-a z^3-5 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+12 a^4 z^2+8 a^2 z^2+2 z^2-4 a^5 z-9 a^3 z-5 a z-a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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          4 -4
4         81 7
2        114  -7
0       158   7
-2      1513    -2
-4     1413     1
-6    1015      5
-8   714       -7
-10  310        7
-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}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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