L11a307

L11a306

L11a308

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Link Presentations

[edit Notes on L11a307's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,5,17,6 X22,14,9,13 X14,18,15,17 X20,7,21,8 X18,22,19,21 X6,15,7,16 X8,19,1,20
Gauss code	{1, -2, 3, -4, 5, -10, 8, -11}, {4, -1, 2, -3, 6, -7, 10, -5, 7, -9, 11, -8, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^3 v^3-3 u^3 v^2+3 u^3 v-3 u^2 v^3+9 u^2 v^2-8 u^2 v+3 u^2+3 u v^3-8 u v^2+9 u v-3 u+3 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+5 z a^5-z^7 a^3-4 z^5 a^3-7 z^3 a^3-5 z a^3+z^5 a+2 z^3 a$ (db)
Kauffman polynomial	$a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-11 a^9 z^5+11 a^9 z^3-7 a^9 z+a^9 z^{-1} +6 a^8 z^8-5 a^8 z^6-3 a^8 z^4+4 a^8 z^2-a^8+4 a^7 z^9+3 a^7 z^7-14 a^7 z^5+12 a^7 z^3-5 a^7 z+a^7 z^{-1} +a^6 z^{10}+13 a^6 z^8-31 a^6 z^6+27 a^6 z^4-9 a^6 z^2+8 a^5 z^9-8 a^5 z^7-8 a^5 z^5+10 a^5 z^3-a^5 z+a^4 z^{10}+13 a^4 z^8-40 a^4 z^6+41 a^4 z^4-16 a^4 z^2+4 a^3 z^9-a^3 z^7-17 a^3 z^5+18 a^3 z^3-4 a^3 z+6 a^2 z^8-16 a^2 z^6+13 a^2 z^4-4 a^2 z^2+4 a z^7-11 a z^5+7 a z^3+z^6-2 z^4$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   4 1   -3 -2                 8 3     5 -4               10 5       -5 -6             9 7         2 -8           10 10           0 -10         7 9             -2 -12       5 10               5 -14     3 7                 -4 -16     5                   5 -18 1 3                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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