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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 L11a145's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X14,7,15,8 X20,13,21,14 X22,15,7,16 X16,21,17,22 X6,18,1,17
Gauss code {1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a145 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+7 t(1) t(2)^3-2 t(2)^3+5 t(1)^2 t(2)^2-9 t(1) t(2)^2+5 t(2)^2-2 t(1)^2 t(2)+7 t(1) t(2)-5 t(2)-2 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial \frac{17}{q^{9/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{5/2}}+q^{3/2}-\frac{12}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{15}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+2 z a^7-z^7 a^5-4 z^5 a^5-6 z^3 a^5-4 z a^5-z^7 a^3-3 z^5 a^3-z^3 a^3+2 z a^3+a^3 z^{-1} +z^5 a+2 z^3 a-z a-a z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-6 z^3 a^9+z a^9-6 z^8 a^8+13 z^6 a^8-13 z^4 a^8+4 z^2 a^8-5 z^9 a^7+10 z^7 a^7-12 z^5 a^7+5 z^3 a^7-z a^7-2 z^{10} a^6-4 z^8 a^6+19 z^6 a^6-28 z^4 a^6+12 z^2 a^6-10 z^9 a^5+29 z^7 a^5-38 z^5 a^5+23 z^3 a^5-5 z a^5-2 z^{10} a^4-4 z^8 a^4+20 z^6 a^4-21 z^4 a^4+8 z^2 a^4-5 z^9 a^3+10 z^7 a^3-4 z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -6 z^8 a^2+16 z^6 a^2-10 z^4 a^2+2 z^2 a^2-a^2-4 z^7 a+11 z^5 a-6 z^3 a-z a+a z^{-1} -z^6+2 z^4 (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 \
4           1-1
2          3 3
0         41 -3
-2        83  5
-4       85   -3
-6      107    3
-8     89     1
-10    79      -2
-12   48       4
-14  27        -5
-16 14         3
-18 2          -2
-201           1
Integral Khovanov Homology

(db, data source)

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