From Knot Atlas
Jump to: navigation, search






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

Visit L11a495 at Knotilus!

Link Presentations

[edit Notes on L11a495's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X20,5,21,6 X18,21,5,22 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code {1, -10, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, -7}
A Braid Representative
A Morse Link Presentation L11a495 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 (u-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-7} +3 q^{-6} -7 q^{-5} +q^4+12 q^{-4} -4 q^3-16 q^{-3} +10 q^2+21 q^{-2} -14 q-20 q^{-1} +19 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 \left(-z^2\right)-a^6+2 a^4 z^4+3 a^4 z^2+a^4-a^2 z^6-a^2 z^4+z^4 a^{-2} +a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -z^6-2 z^4-4 z^2-2 z^{-2} -5 (db)
Kauffman polynomial a^7 z^7-4 a^7 z^5+5 a^7 z^3-2 a^7 z+3 a^6 z^8-11 a^6 z^6+13 a^6 z^4-7 a^6 z^2+2 a^6+4 a^5 z^9-11 a^5 z^7+5 a^5 z^5+5 a^5 z^3-3 a^5 z+2 a^4 z^{10}+5 a^4 z^8-33 a^4 z^6+41 a^4 z^4+z^4 a^{-4} -17 a^4 z^2+2 a^4+12 a^3 z^9-31 a^3 z^7+21 a^3 z^5+4 z^5 a^{-3} -6 a^3 z^3+3 a^3 z+2 a^2 z^{10}+15 a^2 z^8-48 a^2 z^6+10 z^6 a^{-2} +34 a^2 z^4-10 z^4 a^{-2} -a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -6 a^2-4 a^{-2} +8 a z^9-5 a z^7+14 z^7 a^{-1} -10 a z^5-18 z^5 a^{-1} -2 a z^3+4 z^3 a^{-1} +7 a z+3 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +13 z^8-16 z^6-5 z^4+15 z^2+2 z^{-2} -9 (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 \
9           11
7          3 -3
5         71 6
3        73  -4
1       127   5
-1      1211    -1
-3     98     1
-5    712      5
-7   59       -4
-9  27        5
-11 15         -4
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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.