L11a78

L11a77

L11a79

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Link Presentations

[edit Notes on L11a78's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(v^4-4 v^3+7 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+22 q^{3/2}-22 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+6 a z^3-9 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -2 a^3 z+8 a z-9 z a^{-1} +3 z a^{-3} -a^3 z^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-7} +5 z^6 a^{-6} -5 z^4 a^{-6} +10 z^7 a^{-5} -15 z^5 a^{-5} +5 z^3 a^{-5} +10 z^8 a^{-4} +a^4 z^6-10 z^6 a^{-4} -3 a^4 z^4-4 z^4 a^{-4} +3 a^4 z^2+5 z^2 a^{-4} -a^4- a^{-4} +5 z^9 a^{-3} +3 a^3 z^7+11 z^7 a^{-3} -8 a^3 z^5-35 z^5 a^{-3} +8 a^3 z^3+23 z^3 a^{-3} -4 a^3 z-6 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +z^{10} a^{-2} +4 a^2 z^8+18 z^8 a^{-2} -5 a^2 z^6-32 z^6 a^{-2} -5 a^2 z^4+6 z^4 a^{-2} +10 a^2 z^2+12 z^2 a^{-2} -4 a^2-4 a^{-2} +3 a z^9+8 z^9 a^{-1} +5 a z^7+3 z^7 a^{-1} -25 a z^5-36 z^5 a^{-1} +28 a z^3+38 z^3 a^{-1} -15 a z-17 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +z^{10}+12 z^8-23 z^6+3 z^4+14 z^2-7$ (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 14                       1 1 12                     4   -4 10                   6 1   5 8                 9 4     -5 6               11 6       5 4             11 9         -2 2           11 11           0 0         8 13             5 -2       5 9               -4 -4     2 8                 6 -6   1 5                   -4 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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