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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X14,9,15,10 X6718 X20,15,7,16 X16,6,17,5 X4,20,5,19 X10,4,11,3 X12,17,13,18 X18,11,19,12 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 8q-9/2 + 13q-7/2 - 16q-5/2 + 17q-3/2 - 17q-1/2 + 12q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 2q-14 - 4q-12 + q-10 - q-8 - 2q-6 + 4q-4 - q-2 + 7 + q2 + q4 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + 3az-1 + 6az + 7az3 + 4az5 + az7 - a3z-1 - 4a3z - 5a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

a-3z3 - a-3z5 + 5a-2z4 - 4a-2z6 - 2a-1z-1 + 5a-1z - 10a-1z3 + 15a-1z5 - 8a-1z7 + 3 - 7z4 + 13z6 - 8z8 - 3az-1 + 10az - 28az3 + 32az5 - 10az7 - 3az9 + 3a2 - a2z2 - 17a2z4 + 29a2z6 - 15a2z8 - a3z-1 + 7a3z - 23a3z3 + 28a3z5 - 9a3z7 - 3a3z9 + a4 - 3a4z2 + a4z4 + 8a4z6 - 7a4z8 + 2a5z - 5a5z3 + 11a5z5 - 7a5z7 - 2a6z2 + 6a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   3   j = 4                 6 1   j = 2               7 4     j = 0             10 5       j = -2           8 8         j = -4         8 9           j = -6       5 8             j = -8     3 8               j = -10   1 5                 j = -12   3                   j = -14 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, 76]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 76]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 9, 15, 10], X[6, 7, 1, 8], X[20, 15, 7, 16], > X[16, 6, 17, 5], X[4, 20, 5, 19], X[10, 4, 11, 3], X[12, 17, 13, 18], > X[18, 11, 19, 12], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 7, -6, 5, -3}, > {3, -1, 2, -7, 9, -8, 10, -2, 4, -5, 8, -9, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 8 13 16 17 17 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 9 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 4 -10 -8 2 4 -2 2 4 7 + q - q - q + --- - --- + q - q - -- + -- - q + q + q + 14 12 6 4 q q q q 6 8 10 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 76]][a, z]
Out[8]=	3 3 -2 3 a a 2 z 3 5 2 z 3 3 3 --- + --- - -- - --- + 6 a z - 4 a z + a z - ---- + 7 a z - 5 a z + a z z z a a 5 5 3 z 5 3 5 7 > a z - -- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 76]][a, z]
Out[9]=	3 2 4 2 3 a a 5 z 3 5 2 2 3 + 3 a + a - --- - --- - -- + --- + 10 a z + 7 a z + 2 a z - a z - a z z z a 3 3 4 2 6 2 z 10 z 3 3 3 5 3 7 3 > 3 a z - 2 a z + -- - ----- - 28 a z - 23 a z - 5 a z + a z - 3 a a 4 5 5 4 5 z 2 4 4 4 6 4 z 15 z 5 > 7 z + ---- - 17 a z + a z + 6 a z - -- + ----- + 32 a z + 2 3 a a a 6 3 5 5 5 7 5 6 4 z 2 6 4 6 6 6 > 28 a z + 11 a z - a z + 13 z - ---- + 29 a z + 8 a z - 4 a z - 2 a 7 8 z 7 3 7 5 7 8 2 8 4 8 9 > ---- - 10 a z - 9 a z - 7 a z - 8 z - 15 a z - 7 a z - 3 a z - a 3 9 > 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 5 3 8 5 8 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 9 8 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 5 t + 7 q t + 4 q t + 6 q t + q t + 3 q t + 4 2 4 2 q t q t q t 8 4 > q t

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