L11a458

L11a457

L11a459

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Link Presentations

[edit Notes on L11a458's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X20,9,21,10 X18,8,19,7 X22,19,13,20 X8,14,9,13 X12,15,5,16 X10,21,11,22 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w^{3}-2uv^{3}w^{2}+uv^{3}w-2uv^{2}w^{3}+7uv^{2}w^{2}-6uv^{2}w+uv^{2}+uvw^{3}-5uvw^{2}+7uvw-2uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+2v^{2}w^{3}-7v^{2}w^{2}+5v^{2}w-v^{2}-vw^{3}+6vw^{2}-7vw+2v-w^{2}+2w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-14+22q^{-1}-24q^{-2}+26q^{-3}-22q^{-4}+17q^{-5}-11q^{-6}+5q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}+2a^{4}z^{6}+6a^{4}z^{4}+5a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-8a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+2$ (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 2 a^4 z^{10}+2 a^2 z^{10}+8 a^5 z^9+14 a^3 z^9+6 a z^9+13 a^6 z^8+22 a^4 z^8+16 a^2 z^8+7 z^8+11 a^7 z^7+3 a^5 z^7-17 a^3 z^7-5 a z^7+4 z^7 a^{-1} +5 a^8 z^6-18 a^6 z^6-53 a^4 z^6-47 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-15 a^7 z^5-24 a^5 z^5-7 a^3 z^5-8 a z^5-9 z^5 a^{-1} -4 a^8 z^4+6 a^6 z^4+40 a^4 z^4+45 a^2 z^4-2 z^4 a^{-2} +13 z^4+4 a^7 z^3+13 a^5 z^3+12 a^3 z^3+8 a z^3+5 z^3 a^{-1} -a^6 z^2-14 a^4 z^2-20 a^2 z^2+z^2 a^{-2} -6 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed 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 j} , alternation over $r$).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               14 6       8 -3             13 11         -2 -5           13 11           2 -7         9 13             4 -9       8 13               -5 -11     4 10                 6 -13   1 7                   -6 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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