L11a30

L11a29

L11a31

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Link Presentations

[edit Notes on L11a30's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-9 t(1) t(2)^3+13 t(2)^3+13 t(1) t(2)^2-9 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{14}{q^{9/2}}-q^{7/2}+\frac{17}{q^{7/2}}+3 q^{5/2}-\frac{20}{q^{5/2}}-7 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+12 \sqrt{q}-\frac{17}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 (-z)+3 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3-7 a^3 z-4 a^3 z^{-1} -z a^{-3} -a z^5+2 a z^3+2 z^3 a^{-1} +4 a z+3 a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-8 a^3 z^9-4 a z^9-6 a^6 z^8-15 a^4 z^8-15 a^2 z^8-6 z^8-4 a^7 z^7-4 a^5 z^7+a^3 z^7-4 a z^7-5 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+42 a^4 z^6+40 a^2 z^6-3 z^6 a^{-2} +8 z^6+10 a^7 z^5+33 a^5 z^5+37 a^3 z^5+22 a z^5+7 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-4 a^6 z^4-35 a^4 z^4-41 a^2 z^4+5 z^4 a^{-2} -7 z^4-8 a^7 z^3-36 a^5 z^3-52 a^3 z^3-30 a z^3-4 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+a^6 z^2+12 a^4 z^2+17 a^2 z^2-2 z^2 a^{-2} +5 z^2+3 a^7 z+16 a^5 z+25 a^3 z+16 a z+3 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 7 2     -5 0               10 5       5 -2             11 8         -3 -4           9 9           0 -6         8 11             3 -8       6 9               -3 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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