From Knot Atlas
Jump to: navigation, search






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

Visit L11a98 at Knotilus!

Link Presentations

[edit Notes on L11a98's Link Presentations]

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X22,11,5,12 X4,21,1,22 X18,14,19,13 X16,8,17,7 X8,18,9,17 X20,16,21,15 X14,20,15,19
Gauss code {1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 10, -8, 9, -7, 11, -10, 6, -5}
A Braid Representative
A Morse Link Presentation L11a98 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+7 t(2)^4+8 t(1) t(2)^3-9 t(2)^3-9 t(1) t(2)^2+8 t(2)^2+7 t(1) t(2)-5 t(2)-2 t(1)+2}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+10 q^{7/2}-15 q^{5/2}+19 q^{3/2}-22 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -a z^7-z^7 a^{-1} +a^3 z^5-4 a z^5-3 z^5 a^{-1} +z^5 a^{-3} +3 a^3 z^3-7 a z^3-3 z^3 a^{-1} +2 z^3 a^{-3} +3 a^3 z-5 a z-2 z a^{-1} +2 z a^{-3} +a^3 z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +5 a^5 z^3-2 a^5 z+3 a^4 z^8-11 a^4 z^6+10 z^6 a^{-4} +13 a^4 z^4-9 z^4 a^{-4} -5 a^4 z^2+5 z^2 a^{-4} -2 a^{-4} +4 a^3 z^9-11 a^3 z^7+15 z^7 a^{-3} +7 a^3 z^5-21 z^5 a^{-3} +10 z^3 a^{-3} +2 a^3 z-2 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +2 a^2 z^{10}+5 a^2 z^8+14 z^8 a^{-2} -30 a^2 z^6-18 z^6 a^{-2} +34 a^2 z^4-2 z^4 a^{-2} -12 a^2 z^2+10 z^2 a^{-2} +a^2-5 a^{-2} +12 a z^9+8 z^9 a^{-1} -28 a z^7-z^7 a^{-1} +14 a z^5-22 z^5 a^{-1} -2 a z^3+13 z^3 a^{-1} +2 a z-4 z a^{-1} +2 a^{-1} z^{-1} +2 z^{10}+16 z^8-47 z^6+29 z^4-2 z^2-3 (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 \
12           1-1
10          3 3
8         71 -6
6        83  5
4       117   -4
2      118    3
0     1012     2
-2    810      -2
-4   410       6
-6  38        -5
-8 15         4
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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).

Back to the top.