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

Visit L11n305 at Knotilus!

Link Presentations

[edit Notes on L11n305's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X21,15,22,14 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, {11, -2, -3, 9, 5, -7, 6, -8, -4, 3, -9, 4}
A Braid Representative
A Morse Link Presentation L11n305 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^2 t(3)^4+t(1) t(2) t(3)^4-t(2) t(3)^4-t(2)^2 t(3)^3+t(1) t(3)^3-2 t(1) t(2) t(3)^3+2 t(2) t(3)^3-t(3)^3+t(2)^2 t(3)^2-t(1) t(3)^2+2 t(1) t(2) t(3)^2-2 t(2) t(3)^2+t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)-2 t(1) t(2) t(3)+2 t(2) t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -2+4 q^{-1} -6 q^{-2} +9 q^{-3} -8 q^{-4} +9 q^{-5} -6 q^{-6} +5 q^{-7} -2 q^{-8} + q^{-9} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+2 a^8 z^{-2} +3 a^8-3 z^4 a^6-12 z^2 a^6-5 a^6 z^{-2} -15 a^6+2 z^6 a^4+11 z^4 a^4+22 z^2 a^4+4 a^4 z^{-2} +17 a^4-2 z^4 a^2-6 z^2 a^2-a^2 z^{-2} -5 a^2 (db)
Kauffman polynomial z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-6 z^5 a^9+3 z^3 a^9+z a^9+2 z^8 a^8-4 z^6 a^8-4 z^2 a^8-2 a^8 z^{-2} +6 a^8+z^9 a^7+2 z^7 a^7-14 z^5 a^7+19 z^3 a^7-16 z a^7+5 a^7 z^{-1} +6 z^8 a^6-22 z^6 a^6+36 z^4 a^6-36 z^2 a^6-5 a^6 z^{-2} +20 a^6+z^9 a^5+4 z^7 a^5-23 z^5 a^5+43 z^3 a^5-33 z a^5+9 a^5 z^{-1} +4 z^8 a^4-16 z^6 a^4+34 z^4 a^4-32 z^2 a^4-4 a^4 z^{-2} +17 a^4+4 z^7 a^3-15 z^5 a^3+30 z^3 a^3-21 z a^3+5 a^3 z^{-1} +z^6 a^2+2 z^4 a^2-5 z^2 a^2-a^2 z^{-2} +4 a^2+3 z^3 a-5 z a+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 \
1         2-2
-1        2 2
-3       53 -2
-5      41  3
-7     45   1
-9    54    1
-11   25     3
-13  34      -1
-15 14       3
-17 1        -1
-191         1
Integral Khovanov Homology

(db, data source)

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

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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