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Link Presentations

[edit Notes on L11a478's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X16,8,5,7 X18,9,19,10 X22,15,17,16 X14,19,15,20 X20,13,21,14 X12,21,13,22 X8,17,9,18 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {9, -4, 6, -7, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -8, 7, -6, 5, -3}
A Braid Representative
A Morse Link Presentation L11a478 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(v^2 w^2-2 v^2 w+v w^3-3 v w^2+3 v w-v+2 w^2-w\right)}{\sqrt{u} v w^2} (db)
Jones polynomial  q^{-9} -2 q^{-8} +6 q^{-7} -10 q^{-6} +15 q^{-5} -17 q^{-4} +19 q^{-3} -q^2-16 q^{-2} +4 q+13 q^{-1} -8 (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8 z^{-2} +2 a^8-2 z^4 a^6-5 z^2 a^6-2 a^6 z^{-2} -6 a^6+z^6 a^4+2 z^4 a^4+3 z^2 a^4+a^4 z^{-2} +3 a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2+a^2-z^4-z^2 (db)
Kauffman polynomial z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-5 z^5 a^9+2 z^3 a^9+z a^9+3 z^8 a^8-6 z^6 a^8+3 z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+3 z^9 a^7-4 z^7 a^7+4 z^3 a^7-7 z a^7+2 a^7 z^{-1} +z^{10} a^6+8 z^8 a^6-27 z^6 a^6+37 z^4 a^6-27 z^2 a^6-2 a^6 z^{-2} +11 a^6+7 z^9 a^5-9 z^7 a^5-2 z^5 a^5+12 z^3 a^5-9 z a^5+2 a^5 z^{-1} +z^{10} a^4+12 z^8 a^4-32 z^6 a^4+36 z^4 a^4-21 z^2 a^4-a^4 z^{-2} +7 a^4+4 z^9 a^3+4 z^7 a^3-19 z^5 a^3+15 z^3 a^3-2 z a^3+7 z^8 a^2-8 z^6 a^2+7 z^7 a-11 z^5 a+4 z^3 a-z a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 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          3 3
1         51 -4
-1        83  5
-3       107   -3
-5      96    3
-7     810     2
-9    79      -2
-11   38       5
-13  37        -4
-15 15         4
-17 1          -1
-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}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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