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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X20,16,5,15 X16,11,17,12 X18,9,19,10 X10,17,11,18 X8,19,9,20 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 4q-7/2 + 6q-5/2 - 8q-3/2 + 9q-1/2 - 9q1/2 + 7q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-12 - q-8 + q-6 - 2q-4 + 3q4 + 3q8 + q10 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - a-3z3 + a-1z-1 + 2a-1z + 2a-1z3 + a-1z5 + az-1 + 2az + 2az3 + az5 - 2a3z-1 - 4a3z - 2a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-5z3 - 3a-4z4 + a-3z-1 - 3a-3z + 6a-3z3 - 6a-3z5 - a-2 - 2a-2z2 + 9a-2z4 - 7a-2z6 + a-1z-1 - 3a-1z + 2a-1z3 + 8a-1z5 - 6a-1z7 - 2 + 4z2 + 6z6 - 4z8 - az-1 + 10az - 27az3 + 27az5 - 6az7 - az9 - 3a2 + 11a2z2 - 24a2z4 + 22a2z6 - 6a2z8 - 2a3z-1 + 15a3z - 30a3z3 + 18a3z5 - a3z7 - a3z9 - a4 + 5a4z2 - 12a4z4 + 9a4z6 - 2a4z8 - a5z-1 + 5a5z - 8a5z3 + 5a5z5 - a5z7

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 = 10                     1 j = 8                   2   j = 6                 4 1   j = 4               3 2     j = 2             6 4       j = 0           5 5         j = -2         3 4           j = -4       3 5             j = -6     1 3               j = -8   1 3                 j = -10   1                   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, 31]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 31]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[20, 16, 5, 15], > X[16, 11, 17, 12], X[18, 9, 19, 10], X[10, 17, 11, 18], X[8, 19, 9, 20], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 5, -2, 10, -3, 4, -5, 7, -6, > 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 6 8 9 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 7 q - 6 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -8 -6 2 4 8 10 12 14 q + q + --- - q + q - -- + 3 q + 3 q + q - q + q 12 4 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 31]][a, z]
Out[8]=	3 5 3 3 1 1 a 2 a a z 2 z 3 5 z 2 z -(----) + --- + - - ---- + -- - -- + --- + 2 a z - 4 a z + a z - -- + ---- + 3 a z z z z 3 a 3 a a z a a 5 3 3 3 z 5 > 2 a z - 2 a z + -- + a z a
In[9]:=	Kauffman[Link[10, Alternating, 31]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a 3 z 3 z -2 - a - 3 a - a + ---- + --- - - - ---- - -- - --- - --- + 10 a z + 3 a z z z z 3 a a z a 2 3 3 3 3 5 2 2 z 2 2 4 2 z 6 z 2 z > 15 a z + 5 a z + 4 z - ---- + 11 a z + 5 a z - -- + ---- + ---- - 2 5 3 a a a a 4 4 5 3 3 3 5 3 3 z 9 z 2 4 4 4 6 z > 27 a z - 30 a z - 8 a z - ---- + ---- - 24 a z - 12 a z - ---- + 4 2 3 a a a 5 6 8 z 5 3 5 5 5 6 7 z 2 6 4 6 > ---- + 27 a z + 18 a z + 5 a z + 6 z - ---- + 22 a z + 9 a z - a 2 a 7 6 z 7 3 7 5 7 8 2 8 4 8 9 3 9 > ---- - 6 a z - a z - a z - 4 z - 6 a z - 2 a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 3 3 5 5 + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 3 5 4 2 4 4 2 6 2 6 3 8 3 > ----- + - + ---- + 4 q t + 3 q t + 2 q t + 4 q t + q t + 2 q t + 2 2 t 2 q t q t 10 4 > q t

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