L11a124

L11a123

L11a125

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Link Presentations

[edit Notes on L11a124's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,12,17,11 X12,16,13,15 X22,17,5,18 X18,7,19,8 X8,21,9,22 X20,9,21,10 X10,19,11,20 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -7, 8, -9, 3, -4, 11, -2, 4, -3, 5, -6, 9, -8, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(2)^{3}-4t(2)^{3}-9t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-9t(2)-4t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {15}{q^{7/2}}}+{\frac {16}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT 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 -a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -3 z^3 a^7-2 z a^7+z^5 a^5-z^3 a^5-2 z a^5-a^5 z^{-1} +z^5 a^3-z a^3-z^3 a} (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-5z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-2z^{8}a^{10}+z^{6}a^{10}+10z^{4}a^{10}-13z^{2}a^{10}+5a^{10}-2z^{9}a^{9}+z^{7}a^{9}+4z^{5}a^{9}-4z^{3}a^{9}+5za^{9}-2a^{9}z^{-1}-z^{10}a^{8}-3z^{8}a^{8}+7z^{6}a^{8}-2z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-6z^{9}a^{7}+12z^{7}a^{7}-12z^{5}a^{7}+5z^{3}a^{7}-za^{7}-z^{10}a^{6}-8z^{8}a^{6}+22z^{6}a^{6}-23z^{4}a^{6}+7z^{2}a^{6}-a^{6}-4z^{9}a^{5}+2z^{7}a^{5}+4z^{5}a^{5}-2z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-7z^{8}a^{4}+13z^{6}a^{4}-9z^{4}a^{4}+3z^{2}a^{4}-7z^{7}a^{3}+13z^{5}a^{3}-5z^{3}a^{3}+za^{3}-4z^{6}a^{2}+6z^{4}a^{2}-z^{5}a+z^{3}a$ (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 1 2 χ 2                       1 1 0                     3   -3 -2                   5 1   4 -4                 7 4     -3 -6               8 4       4 -8             8 7         -1 -10           8 8           0 -12         5 9             4 -14       4 7               -3 -16     1 5                 4 -18   1 4                   -3 -20   1                     1 -22 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}} $i=-4$ 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 i=-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=-9} ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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 {\mathbb Z}} $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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=-4} 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}^{9}\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} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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=-2} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ 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=1} 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}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ 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 {\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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