L10a45

L10a44

L10a46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a45 at Knotilus!

Link Presentations

[edit Notes on L10a45's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-5uv^{2}+6uv-4u-4v^{3}+6v^{2}-5v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-5q^{9/2}+9q^{7/2}-{\frac {1}{q^{7/2}}}-11q^{5/2}+{\frac {2}{q^{5/2}}}+11q^{3/2}-{\frac {6}{q^{3/2}}}-q^{13/2}+3q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}+z^{5}a^{-3}+2z^{3}a^{-3}+a^{3}z+3za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2az-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-3z^{8}a^{-4}-2z^{8}-3az^{7}-4z^{7}a^{-1}-5z^{7}a^{-3}-4z^{7}a^{-5}-2a^{2}z^{6}+4z^{6}a^{-2}+2z^{6}a^{-4}-3z^{6}a^{-6}-3z^{6}-a^{3}z^{5}+4az^{5}+6z^{5}a^{-1}+10z^{5}a^{-3}+8z^{5}a^{-5}-z^{5}a^{-7}+3a^{2}z^{4}+5z^{4}a^{-2}+4z^{4}a^{-4}+7z^{4}a^{-6}+11z^{4}+3a^{3}z^{3}-az^{3}-3z^{3}a^{-1}-6z^{3}a^{-3}-5z^{3}a^{-5}+2z^{3}a^{-7}-10z^{2}a^{-2}-6z^{2}a^{-4}-4z^{2}a^{-6}-8z^{2}-3a^{3}z+4za^{-1}+2za^{-3}+za^{-5}-a^{2}+5a^{-2}+2a^{-4}+3+a^{3}z^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 3 1   2 8               6 2     -4 6             5 3       2 4           6 6         0 2         5 5           0 0       4 7             3 -2     2 4               -2 -4     4                 4 -6 1 2                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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