L11a15

L11a14

L11a16

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Link Presentations

[edit Notes on L11a15's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X22,16,5,15 X20,18,21,17 X16,22,17,21 X10,20,11,19 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-12 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+11 q^{3/2}-\frac{3}{q^{3/2}}-q^{17/2}+4 q^{15/2}-6 q^{13/2}+9 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^3 a^{-7} + a^{-7} z^{-1} +z^5 a^{-5} +z^3 a^{-5} -z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} -z a^{-3} +z^5 a^{-1} -a z^3+2 z^3 a^{-1} -a z+3 z a^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^7 a^{-9} -3 z^5 a^{-9} +z^3 a^{-9} +4 z^8 a^{-8} -16 z^6 a^{-8} +15 z^4 a^{-8} -z^2 a^{-8} -2 a^{-8} +5 z^9 a^{-7} -20 z^7 a^{-7} +21 z^5 a^{-7} -6 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} -21 z^6 a^{-6} +24 z^4 a^{-6} +z^2 a^{-6} -5 a^{-6} +9 z^9 a^{-5} -33 z^7 a^{-5} +36 z^5 a^{-5} -13 z^3 a^{-5} -z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} -11 z^6 a^{-4} +9 z^4 a^{-4} +2 z^2 a^{-4} -3 a^{-4} +4 z^9 a^{-3} -8 z^7 a^{-3} +9 z^5 a^{-3} -9 z^3 a^{-3} +2 z a^{-3} +4 z^8 a^{-2} -2 z^6 a^{-2} +a^2 z^4-4 z^4 a^{-2} -a^2 z^2+z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} +3 a z^5-4 a z^3-7 z^3 a^{-1} +2 a z+5 z a^{-1} - a^{-1} z^{-1} +4 z^6-3 z^4$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     3   -3 14                   3 1   2 12                 6 3     -3 10               6 3       3 8             7 6         -1 6           7 6           1 4         4 7             3 2       5 7               -2 0     2 6                 4 -2   1 3                   -2 -4   2                     2 -6 1                       -1

Integral Khovanov Homology (db, data source)

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

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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