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Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X14,6,15,5 X18,7,19,8 X16,10,17,9 X10,11,1,12 X6,14,7,13 X4,19,5,20 X20,15,11,16 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 14q-5/2 + 16q-3/2 - 18q-1/2 + 16q1/2 - 13q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 3q-12 + q-10 + 3q-8 - 2q-6 + 5q-4 - 2q-2 + 3 + q2 - q4 + 4q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - 4a-1z - 5a-1z3 - 2a-1z5 + az-1 + 5az + 7az3 + 4az5 + az7 - 2a3z - 2a3z3 - a3z5

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 8a-3z3 + 10a-3z5 - 4a-3z7 - a-2z2 - 6a-2z4 + 13a-2z6 - 6a-2z8 - a-1z-1 + 8a-1z - 28a-1z3 + 33a-1z5 - 7a-1z7 - 3a-1z9 + 1 - 14z4 + 33z6 - 16z8 - az-1 + 10az - 34az3 + 47az5 - 16az7 - 3az9 - 3a2z2 + 3a2z4 + 10a2z6 - 10a2z8 + 4a3z - 13a3z3 + 20a3z5 - 13a3z7 - 3a4z2 + 8a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                     1 j = 8                   3   j = 6                 5 1   j = 4               8 3     j = 2             8 5       j = 0           10 8         j = -2         8 10           j = -4       6 8             j = -6     3 8               j = -8   1 6                 j = -10   3                   j = -12 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 112]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 112]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[16, 10, 17, 9], X[10, 11, 1, 12], X[6, 14, 7, 13], X[4, 19, 5, 20], > X[20, 15, 11, 16], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, > {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 14 16 18 3/2 q - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 13 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 -10 3 2 5 2 2 4 6 8 10 3 - q + --- - --- + q + -- - -- + -- - -- + q - q + 4 q - 2 q + q + 14 12 8 6 4 2 q q q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 112]][a, z]
Out[8]=	3 3 5 1 a z 4 z 3 z 5 z 3 3 3 2 z -(---) + - + -- - --- + 5 a z - 2 a z + -- - ---- + 7 a z - 2 a z - ---- + a z z 3 a 3 a a a a 5 3 5 7 > 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 112]][a, z]
Out[9]=	2 2 1 a 2 z 8 z 3 z z 2 2 4 2 1 - --- - - + --- + --- + 10 a z + 4 a z - -- - -- - 3 a z - 3 a z - a z z 3 a 4 2 a a a 3 3 4 4 8 z 28 z 3 3 3 5 3 4 2 z 6 z 2 4 > ---- - ----- - 34 a z - 13 a z + a z - 14 z + ---- - ---- + 3 a z + 3 a 4 2 a a a 5 5 4 4 6 4 10 z 33 z 5 3 5 5 5 6 > 8 a z - a z + ----- + ----- + 47 a z + 20 a z - 4 a z + 33 z - 3 a a 6 6 7 7 z 13 z 2 6 4 6 4 z 7 z 7 3 7 > -- + ----- + 10 a z - 9 a z - ---- - ---- - 16 a z - 13 a z - 4 2 3 a a a a 8 9 8 6 z 2 8 3 z 9 > 16 z - ---- - 10 a z - ---- - 3 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 6 3 8 6 8 10 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 8 2 2 2 4 2 4 3 6 3 6 4 > ---- + 8 t + 8 q t + 5 q t + 8 q t + 3 q t + 5 q t + q t + 2 q t 8 4 10 5 > 3 q t + q t

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