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Link L11a548.
A graph, L11a548.

Link Presentations

[edit Notes on L11a548's Link Presentations]

Planar diagram presentation X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,20,17,19 X16,22,13,21 X20,16,21,15 X12,17,9,18
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 10, -9}, {11, -3, 8, -10, 9, -8}
A Braid Representative
A Morse Link Presentation L11a548 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 t(2) t(1)+2 t(2) t(3) t(1)-2 t(3) t(1)+t(2) t(4) t(1)-t(2) t(3) t(4) t(1)+2 t(3) t(4) t(1)-2 t(4) t(1)+2 t(2) t(5) t(1)-t(2) t(3) t(5) t(1)+t(3) t(5) t(1)-2 t(2) t(4) t(5) t(1)+t(2) t(3) t(4) t(5) t(1)-2 t(3) t(4) t(5) t(1)+3 t(4) t(5) t(1)-2 t(5) t(1)+2 t(1)+2 t(2)-3 t(2) t(3)+2 t(3)-t(2) t(4)+2 t(2) t(3) t(4)-2 t(3) t(4)+t(4)-2 t(2) t(5)+2 t(2) t(3) t(5)-t(3) t(5)+2 t(2) t(4) t(5)-2 t(2) t(3) t(4) t(5)+2 t(3) t(4) t(5)-2 t(4) t(5)+t(5)-1}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)} \sqrt{t(5)}} (db)
Jones polynomial  q^{-9} - q^{-8} +6 q^{-7} -6 q^{-6} +16 q^{-5} -15 q^{-4} +21 q^{-3} -q^2-15 q^{-2} +5 q+15 q^{-1} -10 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^{10} z^{-2} +a^{10} z^{-4} -7 a^8 z^{-2} -4 a^8 z^{-4} -4 a^8+6 z^2 a^6+15 a^6 z^{-2} +6 a^6 z^{-4} +14 a^6-4 z^4 a^4-10 z^2 a^4-13 a^4 z^{-2} -4 a^4 z^{-4} -16 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+4 a^2 z^{-2} +a^2 z^{-4} +6 a^2-z^4 (db)
Kauffman polynomial z^6 a^{10}-5 z^4 a^{10}+10 z^2 a^{10}+5 a^{10} z^{-2} -a^{10} z^{-4} -10 a^{10}+z^7 a^9-10 z^3 a^9+20 z a^9-15 a^9 z^{-1} +4 a^9 z^{-3} +z^8 a^8+4 z^6 a^8-20 z^4 a^8+30 z^2 a^8+14 a^8 z^{-2} -4 a^8 z^{-4} -25 a^8+z^9 a^7+3 z^7 a^7-30 z^3 a^7+55 z a^7-41 a^7 z^{-1} +12 a^7 z^{-3} +z^{10} a^6+z^8 a^6+9 z^6 a^6-31 z^4 a^6+40 z^2 a^6+18 a^6 z^{-2} -6 a^6 z^{-4} -31 a^6+6 z^9 a^5-8 z^7 a^5+11 z^5 a^5-30 z^3 a^5+55 z a^5-41 a^5 z^{-1} +12 a^5 z^{-3} +z^{10} a^4+10 z^8 a^4-14 z^6 a^4-11 z^4 a^4+30 z^2 a^4+14 a^4 z^{-2} -4 a^4 z^{-4} -25 a^4+5 z^9 a^3-5 z^5 a^3-10 z^3 a^3+20 z a^3-15 a^3 z^{-1} +4 a^3 z^{-3} +10 z^8 a^2-15 z^6 a^2+10 z^2 a^2+5 a^2 z^{-2} -a^2 z^{-4} -10 a^2+10 z^7 a-15 z^5 a+5 z^6-5 z^4+z^5 a^{-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 \
5           1-1
3          4 4
1         61 -5
-1        94  5
-3       1111   0
-5      104    6
-7     511     6
-9    1110      1
-11   515       10
-13  11        0
-15  5         5
-1711          0
-191           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{15}\oplus{\mathbb Z}_2 {\mathbb Z}^{11}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\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

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