L9n28

L9n27

L10a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n28 at Knotilus!
	[edit L9n28 Quick Notes] L9n28 is [math]\displaystyle{ 9^3_{20} }[/math] in the Rolfsen table of links.

[edit L9n28 Further Notes and Views]

Link Presentations

[edit Notes on L9n28's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,15,9 X₈₄₉₃ X_5,17,6,16 X_17,5,18,14 X_10,16,11,15 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -5, 4, -8, 9, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{(u-1) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^5-4 q^4+6 q^3-5 q^2+6 q-4+3 q^{-1} - q^{-2} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+5 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+3 a^{-2} -4 a^{-4} + a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +8 z^6 a^{-2} +5 z^6 a^{-4} +3 z^6+a z^5-z^5 a^{-1} +z^5 a^{-3} +3 z^5 a^{-5} -21 z^4 a^{-2} -13 z^4 a^{-4} -8 z^4-2 a z^3-7 z^3 a^{-1} -8 z^3 a^{-3} -3 z^3 a^{-5} +17 z^2 a^{-2} +18 z^2 a^{-4} +6 z^2 a^{-6} +5 z^2+a z+3 z a^{-1} +7 z a^{-3} +5 z a^{-5} -7 a^{-2} -10 a^{-4} -5 a^{-6} -1-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\		r
	\
j		\

-3

-2

-1

0

1

2

3

4

χ

11

3

9

3

2

-1

7

3

1

2

5

2

3

1

3

4

3

1

2

4

2

-1

1

2

-1

-3

2

-5

1

-1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=1 }[/math]	[math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math]	[math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]

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).

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L9n27

L10a1

L9n28

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Link Presentations

Polynomial invariants

Khovanov Homology

Computer Talk

Modifying This Page

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