L10a163

L10a162

L10a164

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Link Presentations

[edit Notes on L10a163's Link Presentations]

Planar diagram presentation	X8192 X20,10,13,9 X6,20,1,19 X18,7,19,8 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,17,7,18 X2,13,3,14 X14,3,15,4
Gauss code	{1, -9, 10, -5, 6, -3}, {4, -1, 2, -7, 5, -8}, {9, -10, 7, -6, 8, -4, 3, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u w+v w-w+1\right)}{u v w^{3/2}}$ (db)
Jones polynomial	$q^{-7} -4 q^{-6} +8 q^{-5} -12 q^{-4} +q^3+16 q^{-3} -4 q^2-15 q^{-2} +8 q+16 q^{-1} -11$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-8 a^2 z^4+3 z^4+2 a^4 z^2-4 a^2 z^2+2 z^2-a^2+1+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db)
Kauffman polynomial	$a^8 z^4+4 a^7 z^5-2 a^7 z^3+8 a^6 z^6-8 a^6 z^4+3 a^6 z^2+10 a^5 z^7-12 a^5 z^5+4 a^5 z^3+8 a^4 z^8-7 a^4 z^6-3 a^4 z^4+2 a^4 z^2-a^4 z^{-2} +a^4+3 a^3 z^9+9 a^3 z^7-26 a^3 z^5+12 a^3 z^3-a^3 z+2 a^3 z^{-1} +14 a^2 z^8-31 a^2 z^6+z^6 a^{-2} +18 a^2 z^4-2 z^4 a^{-2} -4 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +a^2+3 a z^9+3 a z^7+4 z^7 a^{-1} -20 a z^5-10 z^5 a^{-1} +12 a z^3+6 z^3 a^{-1} -a z+2 a z^{-1} +6 z^8-15 z^6+10 z^4-2 z^2- z^{-2} +1$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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 χ 7                     1 1 5                   3   -3 3                 5 1   4 1               6 3     -3 -1             10 5       5 -3           7 8         1 -5         9 8           1 -7       5 9             4 -9     3 7               -4 -11   1 5                 4 -13   3                   -3 -15 1                     1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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