L11n67

L11n66

L11n68

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Link Presentations

[edit Notes on L11n67's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X9,20,10,21 X13,18,14,19 X7,14,8,15 X17,8,18,9 X19,12,20,13 X15,22,16,5 X21,16,22,17 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -3, 2, -11, 7, -4, 5, -8, 9, -6, 4, -7, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{5 t(1) t(2)^3-t(2)^3-9 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-9 t(2)-t(1)+5}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{17/2}}+\frac{10}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{3}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13} z^{-1} +z^3 a^{11}+2 z a^{11}+a^{11} z^{-1} -z^5 a^9+4 z a^9+2 a^9 z^{-1} -3 z^5 a^7-9 z^3 a^7-7 z a^7-2 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5$ (db)
Kauffman polynomial	$-6 z^4 a^{14}+11 z^2 a^{14}-4 a^{14}-3 z^7 a^{13}+3 z^5 a^{13}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -5 z^8 a^{12}+13 z^6 a^{12}-27 z^4 a^{12}+28 z^2 a^{12}-9 a^{12}-2 z^9 a^{11}-5 z^7 a^{11}+14 z^5 a^{11}-11 z^3 a^{11}+z a^{11}+a^{11} z^{-1} -10 z^8 a^{10}+19 z^6 a^{10}-17 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-2 z^9 a^9-8 z^7 a^9+24 z^5 a^9-24 z^3 a^9+12 z a^9-2 a^9 z^{-1} -5 z^8 a^8+3 z^6 a^8+8 z^4 a^8-7 z^2 a^8+2 a^8-6 z^7 a^7+12 z^5 a^7-12 z^3 a^7+9 z a^7-2 a^7 z^{-1} -3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5$ (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 χ -4                   1 1 -6                 3 1 -2 -8               5     5 -10             6 3     -3 -12           8 5       3 -14         7 7         0 -16       7 7           0 -18     3 7             4 -20   3 7               -4 -22   3                 3 -24 3                   -3

Integral Khovanov Homology (db, data source)

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

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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