L11a58

L11a57

L11a59

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Link Presentations

[edit Notes on L11a58's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+9 t(1) t(2)^3-11 t(2)^3-11 t(1) t(2)^2+9 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -3 z^5 a^3-9 z^3 a^3-11 z a^3-4 a^3 z^{-1} +z^7 a+4 z^5 a+8 z^3 a+6 z a+a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -2 z a^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +4 a^6 z^8-5 a^6 z^6-5 a^6 z^4+10 a^6 z^2-4 a^6+3 a^5 z^9+5 a^5 z^7-25 a^5 z^5+27 a^5 z^3-14 a^5 z+4 a^5 z^{-1} +a^4 z^{10}+12 a^4 z^8-24 a^4 z^6+5 a^4 z^4+12 a^4 z^2-7 a^4+8 a^3 z^9+a^3 z^7-30 a^3 z^5+z^5 a^{-3} +30 a^3 z^3-z^3 a^{-3} -13 a^3 z+4 a^3 z^{-1} +a^2 z^{10}+17 a^2 z^8-33 a^2 z^6+4 z^6 a^{-2} +15 a^2 z^4-5 z^4 a^{-2} +3 a^2 z^2+2 z^2 a^{-2} -4 a^2+5 a z^9+7 a z^7+8 z^7 a^{-1} -26 a z^5-12 z^5 a^{-1} +19 a z^3+7 z^3 a^{-1} -5 a z-2 z a^{-1} +a z^{-1} +9 z^8-11 z^6+3 z^4-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 9 3     -6 0               10 6       4 -2             12 10         -2 -4           10 9           1 -6         7 12             5 -8       6 10               -4 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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