Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)


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

Visit L11a5's page at Knotilus.

Visit L11a5's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a5's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,12,5,11 X20,14,21,13 X14,20,15,19 X12,22,13,21 X18,10,19,9 X2,16,3,15
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 7, -8, 11, -2, 3, -10, 8, -7, 9, -6}
A Braid Representative
A Morse Link Presentation L11a5 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -11 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+12 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-5 q^{13/2}+8 q^{11/2}-10 \sqrt{q}+\frac{6}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} +z^5 a^{-5} -a z^3+z^3 a^{-1} +2 z^3 a^{-5} -z^3 a^{-7} -z a^{-3} +2 z a^{-5} -z a^{-7} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-4} -2 z^{10} a^{-6} -5 z^9 a^{-3} -9 z^9 a^{-5} -4 z^9 a^{-7} -6 z^8 a^{-2} -z^8 a^{-4} +2 z^8 a^{-6} -3 z^8 a^{-8} -6 z^7 a^{-1} +12 z^7 a^{-3} +36 z^7 a^{-5} +17 z^7 a^{-7} -z^7 a^{-9} +9 z^6 a^{-2} +18 z^6 a^{-4} +16 z^6 a^{-6} +13 z^6 a^{-8} -6 z^6-4 a z^5+4 z^5 a^{-1} -13 z^5 a^{-3} -46 z^5 a^{-5} -21 z^5 a^{-7} +4 z^5 a^{-9} -a^2 z^4-4 z^4 a^{-2} -23 z^4 a^{-4} -27 z^4 a^{-6} -15 z^4 a^{-8} +6 z^4+4 a z^3+4 z^3 a^{-1} +8 z^3 a^{-3} +22 z^3 a^{-5} +11 z^3 a^{-7} -3 z^3 a^{-9} +z^2 a^{-2} +6 z^2 a^{-4} +11 z^2 a^{-6} +6 z^2 a^{-8} -2 z a^{-3} -4 z a^{-5} -2 z a^{-7} +1-a z^{-1} - a^{-1} z^{-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). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=1 is the signature of L11a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
18           11
16          2 -2
14         31 2
12        52  -3
10       63   3
8      75    -2
6     76     1
4    57      2
2   57       -2
0  37        4
-2 13         -2
-4 3          3
-61           -1
Integral Khovanov Homology

(db, data source)

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