L11a485

L11a484

L11a486

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Link Presentations

[edit Notes on L11a485's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {4, -1, 2, -5, 8, -9, 6, -4, 11, -2, 3, -10, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	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 \frac{(u-1) (v-1) (w-1)^3 \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}}} (db)
Jones 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 q^{-6} -q^5-6 q^{-5} +5 q^4+13 q^{-4} -11 q^3-20 q^{-3} +19 q^2+28 q^{-2} -26 q-30 q^{-1} +32} (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}-2a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-4a^{2}z^{4}-2z^{4}a^{-2}+2a^{2}z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}+6a^{2}+2a^{-2}+z^{8}+4z^{6}+5z^{4}-2z^{2}-2z^{-2}-6$ (db)
Kauffman polynomial	$a^{6}z^{6}+6a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+2a^{5}z+13a^{4}z^{8}-26a^{4}z^{6}+5z^{6}a^{-4}+12a^{4}z^{4}-4z^{4}a^{-4}+2a^{4}z^{2}-4a^{4}+12a^{3}z^{9}-14a^{3}z^{7}+11z^{7}a^{-3}-8a^{3}z^{5}-13z^{5}a^{-3}+2a^{3}z^{3}+4z^{3}a^{-3}+6a^{3}z+4a^{2}z^{10}+24a^{2}z^{8}+15z^{8}a^{-2}-69a^{2}z^{6}-20z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-12a^{2}-4a^{-2}+24az^{9}+12z^{9}a^{-1}-36az^{7}-5z^{7}a^{-1}-3az^{5}-17z^{5}a^{-1}+10az^{3}+12z^{3}a^{-1}+6az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+4z^{10}+26z^{8}-67z^{6}+38z^{4}+10z^{2}+2z^{-2}-11$ (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 11                       1 -1 9                     4   4 7                   7 1   -6 5                 12 4     8 3               14 7       -7 1             18 12         6 -1           16 18           2 -3         12 14             -2 -5       8 16               8 -7     5 12                 -7 -9   1 8                   7 -11   5                     -5 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ 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=-1} $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{12}$ 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}^{12}} 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} ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=0$ ${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{18}$ 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}^{12}\oplus{\mathbb Z}_2^{14}} 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}^{14}} $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ 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} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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