From Knot Atlas
Jump to: navigation, search






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

Visit L11a247 at Knotilus!

Link Presentations

[edit Notes on L11a247's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,14,21,13 X14,5,15,6 X4,19,5,20 X18,15,19,16 X16,8,17,7 X6,18,7,17 X8,22,1,21
Gauss code {1, -4, 2, -7, 6, -10, 9, -11}, {4, -1, 3, -2, 5, -6, 8, -9, 10, -8, 7, -5, 11, -3}
A Braid Representative
A Morse Link Presentation L11a247 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1)^3 (v-1)^3}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-8 q^{9/2}+13 q^{7/2}-18 q^{5/2}+20 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-8 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} -a^3 z+4 a z-6 z a^{-1} +4 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -6 z^4 a^{-6} +2 z^2 a^{-6} +7 z^7 a^{-5} -11 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} +7 z^8 a^{-4} +a^4 z^6-6 z^6 a^{-4} -2 a^4 z^4-3 z^4 a^{-4} +a^4 z^2+3 z^2 a^{-4} +4 z^9 a^{-3} +4 a^3 z^7+8 z^7 a^{-3} -10 a^3 z^5-28 z^5 a^{-3} +7 a^3 z^3+25 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +z^{10} a^{-2} +6 a^2 z^8+15 z^8 a^{-2} -13 a^2 z^6-32 z^6 a^{-2} +6 a^2 z^4+20 z^4 a^{-2} -2 z^2 a^{-2} +4 a z^9+8 z^9 a^{-1} +2 a z^7-z^7 a^{-1} -24 a z^5-30 z^5 a^{-1} +24 a z^3+35 z^3 a^{-1} -8 a z-12 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+14 z^8-36 z^6+25 z^4-4 z^2-1 (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 \
14           11
12          3 -3
10         51 4
8        83  -5
6       105   5
4      108    -2
2     1110     1
0    812      4
-2   59       -4
-4  38        5
-6 15         -4
-8 3          3
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\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=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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.