L11a243

L11a242

L11a244

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Link Presentations

[edit Notes on L11a243's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{\left(v^2-v+1\right) (u v-u-2 v+1) (u v-2 u-v+1)}{u v^2}$ (db)
Jones polynomial	$q^{9/2}-\frac{9}{q^{9/2}}-4 q^{7/2}+\frac{16}{q^{7/2}}+9 q^{5/2}-\frac{21}{q^{5/2}}-16 q^{3/2}+\frac{24}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+20 \sqrt{q}-\frac{25}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z^3+a^5 z-2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -3 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-4 z^3 a^{-1} +5 a z-3 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+2 a^6 z^2+8 a^5 z^7-11 a^5 z^5+6 a^5 z^3-a^5 z+10 a^4 z^8-13 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} +z^2 a^{-4} -a^4+7 a^3 z^9+a^3 z^7+4 z^7 a^{-3} -16 a^3 z^5-9 z^5 a^{-3} +11 a^3 z^3+7 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a^3 z^{-1} +2 a^2 z^{10}+18 a^2 z^8+7 z^8 a^{-2} -41 a^2 z^6-14 z^6 a^{-2} +26 a^2 z^4+7 z^4 a^{-2} -3 a^2 z^2-3 a^2+13 a z^9+6 z^9 a^{-1} -14 a z^7-3 z^7 a^{-1} -10 a z^5-15 z^5 a^{-1} +12 a z^3+15 z^3 a^{-1} -5 a z-6 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^{10}+15 z^8-39 z^6+24 z^4-2 z^2-3$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 10 3     7 2               11 7       -4 0             14 9         5 -2           11 12           1 -4         10 13             -3 -6       6 11               5 -8     3 10                 -7 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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