L11a54

L11a53

L11a55

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Link Presentations

[edit Notes on L11a54's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,6,15,5 X10,4,11,3 X16,12,17,11 X12,16,13,15 X22,14,5,13 X18,9,19,10 X2,18,3,17 X8,19,9,20
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -7, 8, -4, 7, -6, 10, -9, 11, -2, 3, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 (u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-8 q^{9/2}+\frac{1}{q^{9/2}}+12 q^{7/2}-\frac{3}{q^{7/2}}-16 q^{5/2}+\frac{6}{q^{5/2}}+18 q^{3/2}-\frac{11}{q^{3/2}}-q^{13/2}+4 q^{11/2}-18 \sqrt{q}+\frac{14}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z^5+2 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+a z^3+3 z^3 a^{-1} -z^3 a^{-5} -a^3 z+a z+z a^{-1} -z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -6 z^4 a^{-6} +7 z^7 a^{-5} -12 z^5 a^{-5} +3 z^3 a^{-5} +8 z^8 a^{-4} +a^4 z^6-16 z^6 a^{-4} -3 a^4 z^4+12 z^4 a^{-4} +2 a^4 z^2-4 z^2 a^{-4} +6 z^9 a^{-3} +3 a^3 z^7-11 z^7 a^{-3} -9 a^3 z^5+11 z^5 a^{-3} +7 a^3 z^3-6 z^3 a^{-3} -2 a^3 z+2 z a^{-3} +2 z^{10} a^{-2} +4 a^2 z^8+6 z^8 a^{-2} -9 a^2 z^6-24 z^6 a^{-2} +3 a^2 z^4+29 z^4 a^{-2} -8 z^2 a^{-2} +4 a z^9+10 z^9 a^{-1} -8 a z^7-29 z^7 a^{-1} +4 a z^5+37 z^5 a^{-1} -a z^3-18 z^3 a^{-1} -2 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}+2 z^8-14 z^6+17 z^4-6 z^2-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 7 3     -4 6               9 5       4 4             9 7         -2 2           9 9           0 0         7 11             4 -2       4 7               -3 -4     2 7                 5 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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