L11a6

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Link Presentations

[edit Notes on L11a6's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(2 v^4-4 v^3+5 v^2-4 v+2\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$22 q^{9/2}-22 q^{7/2}+18 q^{5/2}-15 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+4 q^{17/2}-8 q^{15/2}+14 q^{13/2}-19 q^{11/2}+8 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^5 a^{-7} -2 z^3 a^{-7} + a^{-7} z^{-1} +z^7 a^{-5} +3 z^5 a^{-5} +2 z^3 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +3 z^3 a^{-3} +2 z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -6 z^4 a^{-10} +2 z^2 a^{-10} +7 z^7 a^{-9} -10 z^5 a^{-9} +4 z^3 a^{-9} +8 z^8 a^{-8} -9 z^6 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +6 z^9 a^{-7} -2 z^7 a^{-7} -6 z^5 a^{-7} +4 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +11 z^8 a^{-6} -22 z^6 a^{-6} +6 z^4 a^{-6} +9 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -15 z^7 a^{-5} +z^5 a^{-5} +3 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +9 z^8 a^{-4} -21 z^6 a^{-4} +4 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -2 z^7 a^{-3} -14 z^5 a^{-3} +12 z^3 a^{-3} -4 z a^{-3} +6 z^8 a^{-2} -11 z^6 a^{-2} +2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+z^2$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   5 1   4 14                 9 3     -6 12               10 5       5 10             12 9         -3 8           10 10           0 6         8 12             4 4       7 10               -3 2     3 10                 7 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

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). Back to the top.

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