L11a144

L11a143

L11a145

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Link Presentations

[edit Notes on L11a144's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X20,14,21,13 X6,19,1,20 X18,11,19,12 X12,6,13,5 X22,16,7,15 X4,18,5,17 X16,22,17,21
Gauss code	{1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -11, 10, -7, 6, -5, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^4-5 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u-2 v^3+5 v^2-5 v+2}{u v^2}$ (db)
Jones polynomial	$3 q^{9/2}-\frac{4}{q^{9/2}}-6 q^{7/2}+\frac{8}{q^{7/2}}+11 q^{5/2}-\frac{13}{q^{5/2}}-15 q^{3/2}+\frac{16}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+17 \sqrt{q}-\frac{19}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+4 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+a z^3+6 z^3 a^{-1} -3 z^3 a^{-3} -3 a z+4 z a^{-1} -2 z a^{-3} +a^3 z^{-1} -a z^{-1}$ (db)
Kauffman polynomial	$-2 z^{10} a^{-2} -2 z^{10}-6 a z^9-10 z^9 a^{-1} -4 z^9 a^{-3} -10 a^2 z^8-z^8 a^{-2} -3 z^8 a^{-4} -8 z^8-11 a^3 z^7+2 a z^7+28 z^7 a^{-1} +14 z^7 a^{-3} -z^7 a^{-5} -8 a^4 z^6+13 a^2 z^6+18 z^6 a^{-2} +12 z^6 a^{-4} +27 z^6-4 a^5 z^5+15 a^3 z^5+14 a z^5-25 z^5 a^{-1} -16 z^5 a^{-3} +4 z^5 a^{-5} -a^6 z^4+7 a^4 z^4-2 a^2 z^4-22 z^4 a^{-2} -15 z^4 a^{-4} -17 z^4+2 a^5 z^3-7 a^3 z^3-8 a z^3+14 z^3 a^{-1} +9 z^3 a^{-3} -4 z^3 a^{-5} -a^4 z^2-a^2 z^2+9 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2-2 a z-5 z a^{-1} -3 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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                     2   -2 8                   4 1   3 6                 7 2     -5 4               8 4       4 2             9 7         -2 0           10 8           2 -2         7 10             3 -4       6 9               -3 -6     3 8                 5 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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