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 $u$, $v$, $w$, ...)	$-{\frac {5t(1)t(2)^{3}-t(2)^{3}-9t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-9t(2)-t(1)+5}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (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	$-a^{13}z^{-1}+z^{3}a^{11}+2za^{11}+a^{11}z^{-1}-z^{5}a^{9}+4za^{9}+2a^{9}z^{-1}-3z^{5}a^{7}-9z^{3}a^{7}-7za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-6z^{4}a^{14}+11z^{2}a^{14}-4a^{14}-3z^{7}a^{13}+3z^{5}a^{13}-z^{3}a^{13}-za^{13}+a^{13}z^{-1}-5z^{8}a^{12}+13z^{6}a^{12}-27z^{4}a^{12}+28z^{2}a^{12}-9a^{12}-2z^{9}a^{11}-5z^{7}a^{11}+14z^{5}a^{11}-11z^{3}a^{11}+za^{11}+a^{11}z^{-1}-10z^{8}a^{10}+19z^{6}a^{10}-17z^{4}a^{10}+11z^{2}a^{10}-4a^{10}-2z^{9}a^{9}-8z^{7}a^{9}+24z^{5}a^{9}-24z^{3}a^{9}+12za^{9}-2a^{9}z^{-1}-5z^{8}a^{8}+3z^{6}a^{8}+8z^{4}a^{8}-7z^{2}a^{8}+2a^{8}-6z^{7}a^{7}+12z^{5}a^{7}-12z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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   \ -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)

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