L11a343

L11a342

L11a344

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Link Presentations

[edit Notes on L11a343's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,16,19,15 X22,19,11,20 X20,13,21,14 X14,21,15,22 X2,11,3,12 X8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 8, -9, 6, -2, 11, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^9 z+a^9 z^{-1} -3 a^7 z^3-7 a^7 z-5 a^7 z^{-1} +3 a^5 z^5+10 a^5 z^3+14 a^5 z+8 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-8 a^3 z^3-9 a^3 z-4 a^3 z^{-1} +a z^5+2 a z^3+a z$ (db)
Kauffman polynomial	$a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+5 a^{10} z^2-2 a^{10}+4 a^9 z^7-3 a^9 z^5-2 a^9 z^3+a^9 z+a^9 z^{-1} +4 a^8 z^8+3 a^8 z^6-18 a^8 z^4+22 a^8 z^2-9 a^8+3 a^7 z^9+6 a^7 z^7-17 a^7 z^5+16 a^7 z^3-9 a^7 z+5 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-13 a^6 z^6-12 a^6 z^4+29 a^6 z^2-14 a^6+7 a^5 z^9-27 a^5 z^5+33 a^5 z^3-21 a^5 z+8 a^5 z^{-1} +a^4 z^{10}+12 a^4 z^8-26 a^4 z^6+5 a^4 z^4+13 a^4 z^2-8 a^4+4 a^3 z^9+2 a^3 z^7-24 a^3 z^5+25 a^3 z^3-14 a^3 z+4 a^3 z^{-1} +6 a^2 z^8-12 a^2 z^6+3 a^2 z^4+2 a^2 z^2+4 a z^7-10 a z^5+8 a z^3-2 a z+z^6-2 z^4+z^2$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               9 7       -2 -6             11 7         4 -8           8 9           1 -10         8 11             -3 -12       4 8               4 -14     2 8                 -6 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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