L11a443

L11a442

L11a444

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Link Presentations

[edit Notes on L11a443's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^3 w^2-u v^3 w+u v^2 w^3-3 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+3 u v w^2-4 u v w+2 u v-u w^2+2 u w-2 v^3 w^2+v^3 w-2 v^2 w^3+4 v^2 w^2-3 v^2 w+v^2+v w^3-3 v w^2+3 v w-v+w^2-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$-q^8+3 q^7-5 q^6+10 q^5-13 q^4+15 q^3-14 q^2+13 q-9+6 q^{-1} -2 q^{-2} + q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-4 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} -4 z^2+2 a^2-5 a^{-2} +3 a^{-4} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db)
Kauffman polynomial	$z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -7 z^4 a^{-8} +4 z^2 a^{-8} +4 z^7 a^{-7} -7 z^5 a^{-7} +3 z^3 a^{-7} +4 z^8 a^{-6} -5 z^6 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +3 z^9 a^{-5} -3 z^7 a^{-5} +3 z^5 a^{-5} -z^3 a^{-5} +z a^{-5} +z^{10} a^{-4} +4 z^8 a^{-4} -8 z^6 a^{-4} +7 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} -8 z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 z^8 a^{-2} +a^2 z^6-8 z^6 a^{-2} -4 a^2 z^4+9 z^4 a^{-2} +5 a^2 z^2-14 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +2 z^9 a^{-1} +2 a z^7+z^7 a^{-1} -5 a z^5-12 z^5 a^{-1} +2 a z^3+13 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-7 z^6+5 z^4-4 z^2- z^{-2} +3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   3 1   -2 11                 7 2     5 9               8 5       -3 7             7 5         2 5           7 8           1 3         6 7             -1 1       4 8               4 -1     2 5                 -3 -3     4                   4 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

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