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 L11a32's page at Knotilus.

Visit L11a32's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a32's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,8,17,7 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X8,16,9,15 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 7, -8, 9, -3, 4, -5, 8, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L11a32 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(2)^3-6 t(2)^3-9 t(1) t(2)^2+12 t(2)^2+12 t(1) t(2)-9 t(2)-6 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{15/2}-3 q^{13/2}+7 q^{11/2}-11 q^{9/2}+16 q^{7/2}-19 q^{5/2}+18 q^{3/2}-17 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} -2 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} +a^3 z+a z-4 z a^{-1} -z a^{-3} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -8 z^5 a^{-7} +5 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -14 z^6 a^{-6} +15 z^4 a^{-6} -11 z^2 a^{-6} +4 a^{-6} +4 z^9 a^{-5} -5 z^7 a^{-5} -4 z^5 a^{-5} +6 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -42 z^6 a^{-4} +56 z^4 a^{-4} -36 z^2 a^{-4} +9 a^{-4} +8 z^9 a^{-3} -11 z^7 a^{-3} +a^3 z^5-z^5 a^{-3} -2 a^3 z^3+7 z^3 a^{-3} +a^3 z-z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +13 z^8 a^{-2} +3 a^2 z^6-35 z^6 a^{-2} -4 a^2 z^4+38 z^4 a^{-2} +a^2 z^2-19 z^2 a^{-2} +4 a^{-2} +4 z^9 a^{-1} +6 a z^7+3 z^7 a^{-1} -11 a z^5-17 z^5 a^{-1} +11 a z^3+19 z^3 a^{-1} -8 a z-10 z a^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} +6 z^8-5 z^6-4 z^4+5 z^2-2 (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 L11a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
16           1-1
14          2 2
12         51 -4
10        62  4
8       105   -5
6      96    3
4     910     1
2    89      -1
0   510       5
-2  37        -4
-4  5         5
-613          -2
-81           1
Integral Khovanov Homology

(db, data source)

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