L11n298

L11n297

L11n299

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Link Presentations

[edit Notes on L11n298's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{2}w^{3}+3uv^{2}w^{2}-3uv^{2}w+uv^{2}+2uvw^{3}-5uvw^{2}+3uvw-uv-uw^{3}+uw^{2}-v^{2}w^{2}+v^{2}w+vw^{4}-3vw^{3}+5vw^{2}-2vw-w^{4}+3w^{3}-3w^{2}+w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-11q^{3}-2q^{-3}+14q^{2}+7q^{-2}-14q-9q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+z^{2}a^{-4}+a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-2a^{2}z^{2}-5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+3z^{4}+5z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^9 a^{-1} +3 z^9 a^{-3} +14 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+6 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-40 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -23 z^6-11 a z^5-28 z^5 a^{-1} -27 z^5 a^{-3} -10 z^5 a^{-5} +8 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +37 z^4+3 a^3 z^3+14 a z^3+30 z^3 a^{-1} +25 z^3 a^{-3} +6 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 χ 13                   1 1 11                 3   -3 9               5 1   4 7             6 3     -3 5           8 5       3 3         7 7         0 1       7 7           0 -1     4 9             5 -3   3 5               -2 -5   5                 5 -7 2                   -2

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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