L11n434

L11n433

L11n435

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Link Presentations

[edit Notes on L11n434's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X18,11,19,12 X10,17,11,18 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}vw-u^{2}v-u^{2}w+u^{2}+uv^{2}w^{3}-uv^{2}w^{2}+uv^{2}w-uv^{2}-2uvw^{3}+3uvw^{2}-3uvw+2uv+uw^{3}-uw^{2}+uw-u-v^{2}w^{3}+v^{2}w^{2}+vw^{3}-2vw^{2}}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-7q^{3}-q^{-3}+10q^{2}+4q^{-2}-9q-6q^{-1}+9$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}-a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+4z^{2}-5a^{-2}+2a^{-4}+3-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (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 z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +5 z^3 a^{-5} +4 z^8 a^{-4} -13 z^6 a^{-4} +13 z^4 a^{-4} -11 z^2 a^{-4} - a^{-4} z^{-2} +6 a^{-4} +2 z^9 a^{-3} -2 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+8 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +4 a^2 z^4+44 z^4 a^{-2} -4 a^2 z^2-33 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2+12 a^{-2} +2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -4 a z^5-2 z^5 a^{-1} +8 a z^3+10 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-16 z^6+32 z^4-24 z^2- z^{-2} +8} (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                 2   -2 9               4 1   3 7             4 3     -1 5           6 3       3 3         4 5         1 1       5 5           0 -1     2 5             3 -3   2 4               -2 -5   3                 3 -7 1                   -1

Integral Khovanov Homology (db, data source)

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

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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