L11a548

L11a547

L11n1

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Link L11a548.

A graph, L11a548.

Link Presentations

[edit Notes on L11a548's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2t(2)t(1)+2t(2)t(3)t(1)-2t(3)t(1)+t(2)t(4)t(1)-t(2)t(3)t(4)t(1)+2t(3)t(4)t(1)-2t(4)t(1)+2t(2)t(5)t(1)-t(2)t(3)t(5)t(1)+t(3)t(5)t(1)-2t(2)t(4)t(5)t(1)+t(2)t(3)t(4)t(5)t(1)-2t(3)t(4)t(5)t(1)+3t(4)t(5)t(1)-2t(5)t(1)+2t(1)+2t(2)-3t(2)t(3)+2t(3)-t(2)t(4)+2t(2)t(3)t(4)-2t(3)t(4)+t(4)-2t(2)t(5)+2t(2)t(3)t(5)-t(3)t(5)+2t(2)t(4)t(5)-2t(2)t(3)t(4)t(5)+2t(3)t(4)t(5)-2t(4)t(5)+t(5)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}{\sqrt {t(5)}}}}$ (db)
Jones polynomial	$q^{-9}-q^{-8}+6q^{-7}-6q^{-6}+16q^{-5}-15q^{-4}+21q^{-3}-q^{2}-15q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}z^{-4}-7a^{8}z^{-2}-4a^{8}z^{-4}-4a^{8}+6z^{2}a^{6}+15a^{6}z^{-2}+6a^{6}z^{-4}+14a^{6}-4z^{4}a^{4}-10z^{2}a^{4}-13a^{4}z^{-2}-4a^{4}z^{-4}-16a^{4}+z^{6}a^{2}+2z^{4}a^{2}+4z^{2}a^{2}+4a^{2}z^{-2}+a^{2}z^{-4}+6a^{2}-z^{4}$ (db)
Kauffman polynomial	$z^{6}a^{10}-5z^{4}a^{10}+10z^{2}a^{10}+5a^{10}z^{-2}-a^{10}z^{-4}-10a^{10}+z^{7}a^{9}-10z^{3}a^{9}+20za^{9}-15a^{9}z^{-1}+4a^{9}z^{-3}+z^{8}a^{8}+4z^{6}a^{8}-20z^{4}a^{8}+30z^{2}a^{8}+14a^{8}z^{-2}-4a^{8}z^{-4}-25a^{8}+z^{9}a^{7}+3z^{7}a^{7}-30z^{3}a^{7}+55za^{7}-41a^{7}z^{-1}+12a^{7}z^{-3}+z^{10}a^{6}+z^{8}a^{6}+9z^{6}a^{6}-31z^{4}a^{6}+40z^{2}a^{6}+18a^{6}z^{-2}-6a^{6}z^{-4}-31a^{6}+6z^{9}a^{5}-8z^{7}a^{5}+11z^{5}a^{5}-30z^{3}a^{5}+55za^{5}-41a^{5}z^{-1}+12a^{5}z^{-3}+z^{10}a^{4}+10z^{8}a^{4}-14z^{6}a^{4}-11z^{4}a^{4}+30z^{2}a^{4}+14a^{4}z^{-2}-4a^{4}z^{-4}-25a^{4}+5z^{9}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}+20za^{3}-15a^{3}z^{-1}+4a^{3}z^{-3}+10z^{8}a^{2}-15z^{6}a^{2}+10z^{2}a^{2}+5a^{2}z^{-2}-a^{2}z^{-4}-10a^{2}+10z^{7}a-15z^{5}a+5z^{6}-5z^{4}+z^{5}a^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 9 4     5 -3               11 11       0 -5             10 4         6 -7           5 11           6 -9         11 10             1 -11       5 15               10 -13     1 1                 0 -15     5                   5 -17 1 1                     0 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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