L11a513

L11a512

L11a514

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Link Presentations

[edit Notes on L11a513's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X22,18,13,17 X10,20,11,19 X18,10,19,9 X20,12,21,11 X12,22,7,21 X2738 X6,13,1,14
Gauss code	{1, -10, 3, -4, 2, -11}, {10, -1, 7, -6, 8, -9}, {11, -3, 4, -2, 5, -7, 6, -8, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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