L11a85

L11a84

L11a86

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Link Presentations

[edit Notes on L11a85's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^5-4 u v^4+5 u v^3-5 u v^2+4 u v-2 u-2 v^5+4 v^4-5 v^3+5 v^2-4 v+1}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{5}{q^{19/2}}-\frac{8}{q^{17/2}}+\frac{11}{q^{15/2}}-\frac{14}{q^{13/2}}+\frac{12}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z a^{11}-2 a^{11} z^{-1} +3 z^3 a^9+9 z a^9+5 a^9 z^{-1} -3 z^5 a^7-11 z^3 a^7-11 z a^7-3 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+5 z^3 a^5+3 z a^5-z^5 a^3-3 z^3 a^3-2 z a^3$ (db)
Kauffman polynomial	$a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-2 a^{13} z^3+3 a^{12} z^6-2 a^{12} z^4+4 a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-5 a^{11} z+2 a^{11} z^{-1} +4 a^{10} z^8-4 a^{10} z^6-3 a^{10} z^4+10 a^{10} z^2-5 a^{10}+3 a^9 z^9-a^9 z^7-12 a^9 z^5+20 a^9 z^3-14 a^9 z+5 a^9 z^{-1} +a^8 z^{10}+6 a^8 z^8-22 a^8 z^6+15 a^8 z^4+3 a^8 z^2-5 a^8+6 a^7 z^9-14 a^7 z^7-a^7 z^5+13 a^7 z^3-8 a^7 z+3 a^7 z^{-1} +a^6 z^{10}+5 a^6 z^8-27 a^6 z^6+29 a^6 z^4-10 a^6 z^2+3 a^5 z^9-8 a^5 z^7+6 a^5 z^3-a^5 z+3 a^4 z^8-12 a^4 z^6+14 a^4 z^4-5 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-2 a^3 z$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     2   -2 -4                   4 1   3 -6                 6 3     -3 -8               6 3       3 -10             6 6         0 -12           8 6           2 -14         4 7             3 -16       4 7               -3 -18     1 4                 3 -20   1 4                   -3 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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