L11a222

L11a221

L11a223

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Link Presentations

[edit Notes on L11a222's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X22,10,7,9 X20,12,21,11 X10,22,11,21 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

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

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   4 1   -3 16                 6 2     4 14               7 4       -3 12             8 6         2 10           7 7           0 8         6 8             -2 6       4 8               4 4     2 5                 -3 2   1 5                   4 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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