L11a142

L11a141

L11a143

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Link Presentations

[edit Notes on L11a142's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, 9, -10, 8, -7, 4, -2, 11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^4-6 u^2 v^3+6 u^2 v^2-2 u^2 v-2 u v^4+8 u v^3-13 u v^2+8 u v-2 u-2 v^3+6 v^2-6 v+2}{u v^2}$ (db)
Jones polynomial	$4 q^{9/2}-\frac{4}{q^{9/2}}-8 q^{7/2}+\frac{8}{q^{7/2}}+13 q^{5/2}-\frac{14}{q^{5/2}}-18 q^{3/2}+\frac{18}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+20 \sqrt{q}-\frac{21}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} -a z+a^3 z^{-1} -a z^{-1}$ (db)
Kauffman polynomial	$-3 z^{10} a^{-2} -3 z^{10}-9 a z^9-15 z^9 a^{-1} -6 z^9 a^{-3} -13 a^2 z^8-2 z^8 a^{-2} -4 z^8 a^{-4} -11 z^8-12 a^3 z^7+11 a z^7+44 z^7 a^{-1} +20 z^7 a^{-3} -z^7 a^{-5} -8 a^4 z^6+23 a^2 z^6+31 z^6 a^{-2} +14 z^6 a^{-4} +48 z^6-4 a^5 z^5+16 a^3 z^5+7 a z^5-35 z^5 a^{-1} -19 z^5 a^{-3} +3 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-14 a^2 z^4-37 z^4 a^{-2} -14 z^4 a^{-4} -44 z^4+2 a^5 z^3-6 a^3 z^3-8 a z^3+8 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +4 a^2 z^2+12 z^2 a^{-2} +4 z^2 a^{-4} +12 z^2-a^3 z-a z-a^2+a^3 z^{-1} +a z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   5 1   4 6                 8 3     -5 4               10 5       5 2             10 8         -2 0           11 10           1 -2         8 11             3 -4       6 10               -4 -6     3 9                 6 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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