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

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Link Presentations

[edit Notes on L11a342's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,7,17,8 X10,5,1,6 X6374 X4,9,5,10 X20,14,21,13 X22,17,11,18 X18,21,19,22 X14,20,15,19 X2,11,3,12 X8,15,9,16
Gauss code {1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L11a342 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^3 v^2-4 u^3 v+u^3+2 u^2 v^3-10 u^2 v^2+10 u^2 v-2 u^2-2 u v^3+10 u v^2-10 u v+2 u+v^3-4 v^2+2 v}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{20}{q^{9/2}}-\frac{18}{q^{7/2}}+\frac{13}{q^{5/2}}-\frac{8}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{20}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 a^9 z+4 a^9 z^{-1} -6 a^7 z^3-11 a^7 z-6 a^7 z^{-1} +3 a^5 z^5+8 a^5 z^3+10 a^5 z+5 a^5 z^{-1} +a^3 z^5-a^3 z^3-4 a^3 z-2 a^3 z^{-1} -a z^3-a z (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+8 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-5 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^9+5 a^9 z^7-27 a^9 z^5+31 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^{10}+11 a^8 z^8-22 a^8 z^6+4 a^8 z^4+7 a^8 z^2-3 a^8+7 a^7 z^9+4 a^7 z^7-41 a^7 z^5+52 a^7 z^3-29 a^7 z+6 a^7 z^{-1} +a^6 z^{10}+14 a^6 z^8-28 a^6 z^6+15 a^6 z^4-a^6+4 a^5 z^9+8 a^5 z^7-33 a^5 z^5+41 a^5 z^3-23 a^5 z+5 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+5 a^4 z^4-a^4+6 a^3 z^7-10 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z (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 \
2           11
0          2 -2
-2         61 5
-4        83  -5
-6       105   5
-8      108    -2
-10     1010     0
-12    811      3
-14   59       -4
-16  28        6
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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).

See/edit the Link_Splice_Base (expert).

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