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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X15,1,16,4 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11

Gauss Code:

{{1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}}

Jones Polynomial:

- q-7 + 2q-6 - 3q-5 + 5q-4 - 5q-3 + 6q-2 - 5q-1 + 5 - 2q + 2q2

A2 (sl(3)) Invariant:

- q-22 - q-16 + 2q-14 + q-12 + q-10 + 2q-8 + q-6 + 3q-4 + 3q-2 + 4 + 5q2 + 2q4 + 3q6 + 2q8

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 - 2z-2 - 4 - 3z2 + a2z-2 + a2 + a2z2 + a2z4 + 2a4 + 2a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 3a-2z2 - 2a-1z-1 + 4a-1z - a-1z3 + a-1z5 + 2z-2 - 7 + 10z2 - 5z4 + 2z6 - 2az-1 + 6az - 2az3 - 3az5 + 2az7 + a2z-2 - 2a2 - a2z2 + 3a2z4 - 5a2z6 + 2a2z8 + 2a3z3 - 4a3z5 - a3z7 + a3z9 + 4a4 - 16a4z2 + 23a4z4 - 17a4z6 + 4a4z8 - 4a5z + 10a5z3 - 5a5z5 - 2a5z7 + a5z9 + 2a6 - 8a6z2 + 15a6z4 - 10a6z6 + 2a6z8 - 2a7z + 7a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   2 j = 3                 1 1 j = 1               4 1   j = -1             3 3     j = -3           3 2       j = -5         3 4         j = -7       2 2           j = -9     1 3             j = -11   1 2               j = -13   1                 j = -15 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 286]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 286]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[15, 1, 16, 4], X[21, 18, 22, 19], X[9, 21, 10, 20], > X[19, 5, 20, 10], X[11, 16, 12, 17], X[17, 22, 18, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, > {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 3 5 5 6 5 2 5 - q + -- - -- + -- - -- + -- - - - 2 q + 2 q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -16 2 -12 -10 2 -6 3 3 2 4 6 4 - q - q + --- + q + q + -- + q + -- + -- + 5 q + 2 q + 3 q + 14 8 4 2 q q q q 8 > 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 286]][a, z]
Out[8]=	2 2 2 4 6 2 1 a 2 2 2 4 2 6 2 -4 + -- + a + 2 a - a - -- + ----- + -- - 3 z + a z + 2 a z - a z + 2 2 2 2 2 a z a z z 2 4 4 4 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 286]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 4 z -7 - -- - 2 a + 4 a + 2 a + -- + ----- + -- - --- - --- + --- + 6 a z - 2 2 2 2 2 a z z a a z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 z 3 > 4 a z - 2 a z + 10 z + ---- - a z - 16 a z - 8 a z - -- - 2 a z + 2 a a 5 3 3 5 3 7 3 4 2 4 4 4 6 4 z > 2 a z + 10 a z + 7 a z - 5 z + 3 a z + 23 a z + 15 a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 > 3 a z - 4 a z - 5 a z - 5 a z + 2 z - 5 a z - 17 a z - 6 6 7 3 7 5 7 7 7 2 8 4 8 6 8 > 10 a z + 2 a z - a z - 2 a z + a z + 2 a z + 4 a z + 2 a z + 3 9 5 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 2 1 3 2 2 - + 4 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 3 4 3 2 3 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + q t + q t + q t + 2 q t 5 3 5 2 3 2 3 q t q t q t q t q t

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