L11a48

L11a47

L11a49

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Link Presentations

[edit Notes on L11a48's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-4v+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-10q^{9/2}+15q^{7/2}-20q^{5/2}+24q^{3/2}-23{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4az^{3}-4z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+az-2za^{-1}+3za^{-3}-za^{-5}+a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+9z^{7}a^{-5}-15z^{5}a^{-5}+10z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+11z^{8}a^{-4}+a^{4}z^{6}-19z^{6}a^{-4}-2a^{4}z^{4}+14z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+a^{-4}+7z^{9}a^{-3}+4a^{3}z^{7}-10a^{3}z^{5}-19z^{5}a^{-3}+8a^{3}z^{3}+23z^{3}a^{-3}-a^{3}z-13za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+16z^{8}a^{-2}-11a^{2}z^{6}-38z^{6}a^{-2}+3a^{2}z^{4}+26z^{4}a^{-2}+a^{2}z^{2}-6z^{2}a^{-2}+a^{2}+3a^{-2}+5az^{9}+12z^{9}a^{-1}-az^{7}-14z^{7}a^{-1}-16az^{5}-9z^{5}a^{-1}+14az^{3}+18z^{3}a^{-1}-3az-10za^{-1}-az^{-1}+a^{-1}z^{-1}+2z^{10}+11z^{8}-27z^{6}+13z^{4}-2z^{2}+2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   7 1   6 8                 8 3     -5 6               12 7       5 4             12 8         -4 2           11 12           -1 0         10 14             4 -2       5 9               -4 -4     3 10                 7 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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