L11a154

L11a153

L11a155

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Link Presentations

[edit Notes on L11a154's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X22,16,7,15 X14,5,15,6 X4,13,5,14 X20,18,21,17 X12,20,13,19 X18,12,19,11 X16,22,17,21 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 8, -7, 5, -4, 3, -9, 6, -8, 7, -6, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-6t(1)t(2)+2t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {5}{q^{9/2}}}-3q^{7/2}+{\frac {8}{q^{7/2}}}+6q^{5/2}-{\frac {12}{q^{5/2}}}-9q^{3/2}+{\frac {13}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+12{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-2z^{3}a^{3}-2za^{3}-2z^{5}a-5z^{3}a-5za-2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}+za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-3a^{4}z^{8}-4a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}-3a^{5}z^{7}+7a^{3}z^{7}+18az^{7}+5z^{7}a^{-1}-3z^{7}a^{-3}-2a^{6}z^{6}+4a^{4}z^{6}+17a^{2}z^{6}+11z^{6}a^{-2}-z^{6}a^{-4}+23z^{6}-a^{7}z^{5}+6a^{5}z^{5}-12a^{3}z^{5}-27az^{5}+z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-a^{4}z^{4}-28a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}-34z^{4}+3a^{7}z^{3}-5a^{5}z^{3}+7a^{3}z^{3}+21az^{3}-6z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}+15a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-4}+23z^{2}-2a^{7}z+4a^{5}z-a^{3}z-10az-2za^{-1}+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 5 2     3 2               7 4       -3 0             7 5         2 -2           7 8           1 -4         5 6             -1 -6       3 7               4 -8     2 5                 -3 -10     3                   3 -12 1 2                     -1 -14 1                       1

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}} $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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=5} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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